INTEGRABLE CLUSTER DYNAMICS OF DIRECTED … · arxiv:1406.1883v1 [math.ds] 7 jun 2014 integrable cluster dynamics of directed networks and pentagram maps michael gekhtman, …

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INTEGRABLE CLUSTER DYNAMICS OF DIRECTED

NETWORKS AND PENTAGRAM MAPS

MICHAEL GEKHTMAN, MICHAEL SHAPIRO, SERGE TABACHNIKOV,AND ALEK VAINSHTEIN

To the memory of Andrei Zelevinsky

Abstract. The pentagram map was introduced by R. Schwartz more than 20years ago. In 2009, V. Ovsienko, R. Schwartz and S. Tabachnikov establishedLiouville complete integrability of this discrete dynamical system. In 2011, M.Glick interpreted the pentagram map as a sequence of cluster transformationsassociated with a special quiver. Using compatible Poisson structures in clusteralgebras and Poisson geometry of directed networks on surfaces, we generalizeGlick’s construction to include the pentagram map into a family of discreteintegrable maps and we give these maps geometric interpretations.

Contents

1. Introduction 12. Generalized Glick’s quivers and the

(p,q)-dynamics 43. Weighted directed networks and the (x,y)-dynamics 73.1. Weighted directed networks on surfaces 73.2. The (x,y)-dynamics 104. Poisson structure and complete integrability 164.1. Cuts, rims, and conjugate networks 164.2. Poisson structure 204.3. Conserved quantities 234.4. Lax representations 254.5. Complete integrability 274.6. Spectral curve 295. Geometric interpretation 305.1. The case k ≥ 3 305.2. The case k = 2: leapfrog map and circle pattern 38References 44

1. Introduction

The pentagram map was introduced by R. Schwartz more than 20 years ago [34].The map acts on plane polygons by drawing the “ short” diagonals that connectsecond-nearest vertices of a polygon and forming a new polygon, whose vertices aretheir consecutive intersection points, see Fig. 1. The pentagram map commutes

1

http://arxiv.org/abs/1406.1883v1

2 M. GEKHTMAN, M. SHAPIRO, S. TABACHNIKOV, AND A. VAINSHTEIN

with projective transformations, and therefore acts on the projective equivalenceclasses of polygons in the projective plane.

Figure 1. Pentagram map

In fact, the pentagrammap acts on a larger class of twisted polygons. A twisted n-gon is an infinite sequence of points Vi ∈ RP2 such that Vi+n =M(Vi) for all i ∈ Z

and a fixed projective transformation M , called the monodromy. The projectivegroup PGL(3,R) naturally acts on twisted polygons. A polygon is closed if themonodromy is the identity.

Denote by Pn the moduli space of projective equivalence classes of twisted n-gons, and by Cn its subspace consisting of closed polygons. Then Pn and Cn arevarieties of dimensions 2n and 2n − 8, respectively. Denote by T : Pn → Pn

the pentagram map (the ith vertex of the image is the intersection of diagonals(Vi, Vi+2) and (Vi+1, Vi+3).)

One can introduce coordinates X1, Y1, . . . , Xn, Yn in Pn where Xi, Yi are theso-called corner invariants associated with ith vertex, discrete versions of projec-tive curvature, see [36]. In these coordinates, the pentagram map is a rationaltransformation

(1.1) X∗i = Xi

1−Xi−1 Yi−1

1−Xi+1 Yi+1, Y ∗

i = Yi+11−Xi+2 Yi+2

1−Xi Yi

(the indices are taken mod n).In [35], Schwartz proved that the pentagram map was recurrent, and in [36], he

proved that the pentagram map had 2[n/2] + 2 independent integrals, polynomialin the variables Xi, Yi. He conjectured that the pentagram map was a discretecompletely integrable system.

This was proved in [29, 30]: the space Pn has a T -invariant Poisson structurewhose corank equals 2 or 4, according as n is odd or even, and the integrals arein involution. This provides Liouville integrability of the pentagram map on thespace of twisted polygons.

F. Soloviev [40] established algebraic-geometric integrability of the pentagrammap by constructing its Lax (zero curvature) representation. His approach estab-lished complete integrability of the pentagram map on the space of closed polygonsCn as well; a different proof of this result was given in [31].

It is worth mentioning that the continuous limit as n → ∞ of the pentagrammap is the Boussinesq equation, one of the best known completely integrable PDEs.More specifically, in the limit, a twisted polygon becomes a parametric curve (withmonodromy) in the projective plane, and the map becomes a flow on the modulispace of projective equivalence classes of such curve. This flow is identified with

INTEGRABLE CLUSTER DYNAMICS AND PENTAGRAM MAPS 3

the Boussinesq equation, see [34, 30]. Thus the pentagram map is a discretization,both space- and time-wise, of the Boussinesq equation.

R. Schwartz and S. Tabachnikov discovered several configuration theorems ofprojective geometry related to the pentagram map in [38] and found identitiesbetween the integrals of the pentagram map on polygons inscribed into a conicin [39]. R. Schwartz [37] proved that the integrals of the pentagram map do notchange in the 1-parameter family of Poncelet polygons (polygons inscribed into aconic and circumscribed about a conic).

It was shown in [36] that the pentagram map was intimately related to the so-called octahedral recurrence (also known as the discrete Hirota equation), and itwas conjectured in [29, 30] that the pentagram map was related to cluster trans-formations. This relation was discovered and explored by Glick [15] who provedthat the pentagram map, acting on the quotient space Pn/R

∗ (the action of R∗

commutes with the map and is given by the formula Xi 7→ tXi, Yi 7→ t−1Yi), is de-scribed by coefficient dynamics [9] – also known as τ -transformations, see Chapter4 in [11] – for a certain cluster structure.

In this paper, expanding on the research announcement [14], we generalize Glick’swork by including the pentagrammap into a family of discrete completely integrablesystems. Our main tool is Poisson geometry of weighted directed networks onsurfaces. The ingredients necessary for complete integrability – invariant Poissonbrackets, integrals of motion in involution, Lax representation – are recovered fromcombinatorics of the networks.

A. Postnikov [32] introduced such networks in the case of a disk and investigatedtheir transformations and their relation to cluster transformations; most of hisresults are local, and hence remain valid for networks on any surface. Poissonproperties of weighted directed networks in a disk and their relation to r-matrixstructures on GLn are studied in [10]. In [12] these results were further extended tonetworks in an annulus and r-matrix Poisson structures on matrix-valued rationalfunctions. Applications of these techniques to the study of integrable systems canbe found in [13]. A detailed presentation of the theory of weighted directed networksfrom a cluster algebra perspective can be found in Chapters 8–10 of [11].

Our integrable systems, Tk, depend on one discrete parameter k ≥ 2. Thegeometric meaning of k − 1 is the dimension of the ambient projective space. Thecase k = 3 corresponds to the pentagram map, acting on planar polygons.

For k ≥ 4, we interpret Tk as a transformation of a class of twisted polygons inRPk−1, called corrugated polygons. The map is given by intersecting consecutivediagonals of combinatorial length k − 1 (i.e., connecting vertex Vi with Vi+k−1);corrugated polygons are defined as the ones for which such consecutive diagonalsare coplanar. The map Tk is closely related with a pentagram-like map in the plane,involving deeper diagonals of polygons.

For k = 2, we give a different geometric interpretation of our system: the mapT2 acts on pairs of twisted polygons in RP1 having the same monodromy (thesepolygons may be thought of as ideal polygons in the hyperbolic plane by identifyingRP1 with the circle at infinity), and the action is given by an explicit constructionthat we call the leapfrog map whereby one polygon “jumps” over another, see adescription in Section 5. If the ground field is C, we interpret the map T2 in termsof circle patterns studied by O. Schramm [33, 4].


The pentagram map is coming of age, and we finish this introduction by brieflymentioning, in random order, some related work that appeared since our initialresearch announcement [14] was written.

• A variety of multi-dimensional versions of the pentagram map, integrableand non-integrable, was studied by B. Khesin and F. Soloviev [21, 22, 23,24], and by G. Mari-Beffa [25, 26]. The continuous limits of these maps areidentified with the Adler-Gelfand-Dikii flows.

• V. Fock and A. Marshakov [8] described a class of integrable systems onPoisson submanifolds of the affine Poisson-Lie groups PGL(N). The pen-tagram map is a particular example. The quotient of the correspondingintegrable system by the scaling action (see p-dynamics Tk defined in Sec-tion 2) coincides with the integrable system constructed by A. Goncharovand R. Kenyon out of dimer models on a two-dimensional torus and classi-fied by the Newton polygons [18].

• M. Glick [16, 17] established the singularity confinement property of thepentagram map and some other discrete dynamical systems.

• R. Kedem and P. Vichitkunakorn [20] interpreted the pentagram map interms of T -systems.

• The pentagram map is amenable for tropicalization. A study of the tropicallimit of the pentagram map was done by T. Kato [19].

2. Generalized Glick’s quivers and the

(p,q)-dynamics

For any integer n ≥ 2, let p = (p1, . . . , pn) and q = (q1, . . . , qn) be independentvariables. Fix an integer k, 2 ≤ k ≤ n, and consider the quiver (an orientedmultigraph without loops and cycles of length two) Qk,n defined as follows: Qk,n

is a bipartite graph on 2n vertices labeled p1, . . . , pn and q1, . . . , qn (the labelingis cyclic, so that n + 1 is the same as 1). The graph is invariant under the shifti 7→ i+1. Each vertex has two incoming and two outgoing edges. The number k isthe “span” of the quiver, that is, the distance between two outgoing edges from ap-vertex, see Fig. 2 where r = ⌊k/2⌋ − 1 and r + r′ = k − 2 (in other words, r′ = rfor k even and r′ = r + 1 for k odd). For k = 3, we have Glick’s quiver [15].

pi

qi

pi−r’−1 pi−r’ pi+r pi+r+1

qi−r−1 qi−r qi+r’ qi+r’+1

Figure 2. The quiver Qk,n

Let us equip the (p,q)-space with a Poisson structure ·, ·k as follows. Denoteby A = (aij) the 2n× 2n skew-adjacency matrix of Qk,n, assuming that the first nrows and columns correspond to p-vertices. Then we put vi, vjk = aijvivj , wherevi = pi for 1 ≤ i ≤ n and vi = qi−n for n+ 1 ≤ i ≤ 2n.


Consider a transformation of the (p,q)-space to itself denoted by T k and definedas follows (the new variables are marked by asterisk):

(2.1) q∗i =1

pi+r−r′, p∗i = qi

(1 + pi−r′−1)(1 + pi+r+1)pi−r′pi+r

(1 + pi−r′)(1 + pi+r).

Theorem 2.1. (i) The Poisson structure ·, ·k is invariant under the map T k.

(ii) The function∏n

i=1 piqi is an integral of the map T k. Besides, it is Casimir,

and hence the Poisson structure and the map descend to level hypersurfaces of∏ni=1 piqi.

Proof. (i) Recall that given an arbitrary quiverQ, itsmutation at vertex v is definedas follows:

1) for any pair of edges u′ → v, v → u′′ in Q, the edge u′ → u′′ is added;2) all edges incident to v reverse their direction;3) all cycles of length two are erased.

The obtained quiver is said to be mutationally equivalent to the initial one. Assumethat an independent variable τv is assigned to each vertex of Q. According toLemma 4.4 of [11], the cluster transformation of τ -coordinates corresponding tothe quiver mutation at vertex v (also known as cluster Y -dynamics) is defined asfollows:

(2.2) τ∗v =1

τv, τ∗u =

τu(1 + τv)#(u,v) if #(u, v) > 0,

τuτ#(v,u)v

(1+τv)#(v,u) if #(v, u) > 0,

τu otherwise,

where #(u′, u′′) is the number of edges from u′ to u′′ in Q. Note that at most one ofthe numbers #(u, v) and #(v, u) is nonzero for any vertex u. The cluster structureassosiated with the initial quiver Q and initial set of variables τvv∈Q consists ofall quivers mutationally equivalent to Q and of the corresponding sets of variablesobtained by repeated application of (2.2).

Consider the cluster structure associated with the quiver Qk,n. Choose vari-ables p = (p1, . . . , pn) and q = (q1, . . . , qn) as τ -coordinates, and consider clustertransformations corresponding to the quiver mutations at the p-vertices. Thesetransformations commute, and we perform them simultaneously. By (2.2), thisleads to the transformation

(2.3) p∗i =1

pi, q∗i = qi

(1 + pi−r′−1)(1 + pi+r+1)pi−r′pi+r

(1 + pi−r′)(1 + pi+r).

The resulting quiver is identical to Qk,n with the letters p and q interchanged.Indeed, the mutation at pi generates four new edges qi−r → qi+r′+1, qi+r′ →qi+r′+1, qi−r → qi−r−1, and qi+r′ → qi−r−1. The first of them disappears afterthe mutation at pi+1, the second after the mutation at pi+k−1, the third afterthe mutation at pi−k+1, and the fourth after the mutation at pi−1. Therefore,the result of mutations at all p-vertices is just the reversal of all edges of Qk,n.Thus we compose transformation (2.3) with the transformation given by pi = qi,qi = pi+r−r′ and arrive at the transformation T k defined by (2.1). The differencein the formulas for the odd and even k is due to the asymmetry between left andright in the enumeration of vertices in Fig. 2 for odd k, when r′ 6= r.

A Poisson structure ·, · is said to be compatible with a cluster structure ifxi, xj = cijxixj for any two variables from the same cluster, where the constants


cij depend on the cluster (the cluster basis is related to the τ -basis described abovevia monomial transformations; we will not need the explicit description of thesetransformations here). By Theorem 4.5 in [11], the Poisson structure ·, ·k iscompatible with the above cluster structure. Consequently, ·, ·k can be writtenin the basis (2.3) in the same way as above via the adjacency matrix of the resultingquiver. After the vertices are renamed, we arrive back at Qk,n, which means that

·, ·k is invariant under T k.(ii) Invariance of the function

∏ni=1 piqi means the equality

∏ni=1 p

∗i q

∗i =

∏ni=1 piqi,

which is checked directly by inspection of formulas (2.1). The statement that∏ni=1 piqi is Casimir (or, equivalently, commutes with any pi and qi) follows from

the form of the quiver, since every vertex has an equal number (2, exactly) of in-coming and outgoing edges. Hence, the level hypersurface

∏ni=1 piqi = const is a

Poisson submanifold, and, moreover, T k preserves the hypersurface.

Along with the p-dynamics T k, when the mutations are performed at the p-

vertices of the quiver Qk,n, one may consider the respective q-dynamics Tk, when

the mutations are performed at q-vertices. Let us define an auxiliary map Dk givenby

(2.4) pi =1

qi, qi =

1

pi+r−r′.

Note that Dk is almost an involution: D2

k = Sr−r′, where St is the shift by t inindices. The following proposition describes relations between transformations T ,

T−1

, and T.

Proposition 2.2. (i) Transformation Tk coincides with T

−1

k and is given by

(2.5) p∗i =1

qi−r+r′, q∗i = pi

(1 + qi−r)(1 + qi+r′)qi−r−1qi+r′+1

(1 + qi−r−1)(1 + qi+r′+1).

(ii) Transformations Tk and T k are almost conjugated by Dk:

(2.6) Sr−r′ Tk Dk = Dk T k.

(iii) Let Dk,n be given by pi = qi−⌊(n+r−r′)/2⌋, qi = pi. Then

Tk = Dk,n Tn+2−k Dk,n.

Proof. (i) Recall that T k is defined as the composition of the cluster transforma-tion (2.3) and the shift pi = qi, qi = pi+r−r′ Equivalently, we can write T k =Ck Dk, where Ck is given by expressions reciprocal to those in the right-hand side

of (2.3). It is easy to check that Ck is an involution, and that D−1

k = Sr′−r Dk

is given by pi = 1/qi−r+r′, qi = 1/pi. Consequently, T−1

k = D−1

k Ck is givenby (2.5).

To get the same relations for the transformartion Tk one has to use an analog

of (2.3)

q∗i =1

qi, p∗i = pi

(1 + qi−r)(1 + qi+r′)qi−r−1qi+r′+1

(1 + qi−r−1)(1 + qi+r′+1)

and compose it with the map qi = pi, pi = qi−r+r′ , see Fig. 2.(ii) Follows immediately from (i).


(iii) Follows from the fact that Qn+2−k,n locally at the vertex qi+⌊(n+r−r′)/2⌋ hasthe same structure as Qk,n at the vertex pi. This is illustrated in Fig. 3.

pi pi pipi

qi qi

k=3 k=2 k=n k=n−1

qi+m qi+m

Figure 3. The quivers Qk,n for n = 2m and various values of k

For a formal proof note that the values r and r′ corresponding to k = n+ 2− kare given by

(2.7) r = ⌊(n+ r − r′)/2⌋ − r − 1, r + r′ = n− k.

By Theorem 2.1(ii), T k restricts to any hypersurface∏n

i=1 piqi = c. We denote

this restriction by T(c)

k . In what follows, we shall be concerned only with T(1)

k . Note

that T(1)

3 is the pentagram map on Pn/R∗ considered by Glick [15].

3. Weighted directed networks and the (x,y)-dynamics

3.1. Weighted directed networks on surfaces. We start with a very brief de-scription of the theory of weighted directed networks on surfaces with a boundary,adapted for our purposes; see [32, 11] for details. In this paper, we will only needto consider acyclic graphs on a cylinder (equivalently, annulus) C that we positionhorizontally with one boundary circle on the left and another on the right.

Let G be a directed acyclic graph with the vertex set V and the edge set Eembedded in C. G has 2n boundary vertices , each of degree one: n sources on theleft boundary circles and n sinks on the right boundary circle. All the internalvertices of G have degree 3 and are of two types: either they have exactly oneincoming edge (white vertices), or exactly one outgoing edge (black vertices). Toeach edge e ∈ E we assign the edge weight we ∈ R \ 0. A perfect network N isobtained from G by adding an oriented curve ρ without self-intersections (called acut) that joins the left and the right boundary circles and does not contain verticesof G. The points of the space of edge weights EN can be considered as copies of Nwith edges weighted by nonzero real numbers.

Assign an independent variable λ to the cut ρ. The weight of a directed path Pbetween a source and a sink is defined as a signed product of the weights of all edgesalong the path times λd, where d is the intersection index of ρ and P (we assumethat all intersection points are transversal, in which case the intersection index isthe number of intersection points counted with signs). The sign is defined by therotation number of the loop formed by the path, the cut, and parts of the boundarycycles (see [12] for details). In particular, the sign of a simple path going from oneboundary circle to the other one and intersecting the cut d times in the samedirection equals (−1)d. Besides, if a path P can be decomposed in a path P ′ and a


simple cycle, then the signs of P and P ′ are opposite. The boundary measurement

between a given source and a given sink is then defined as the sum of path weightsover all (not necessary simple) paths between them. A boundary measurement isrational in the weights of edges and λ, see Proposition 2.2 in [12]; in particular,if the network does not have oriented cycles then the boundary measurements arepolynomials in edge weights, λ and λ−1.

Boundary measurements are organized in a boundary measurement matrix, thusgiving rise to the boundary measurement map from EN to the space of n×n rationalmatrix functions. The gauge group acts on EN as follows: for any internal vertexv of N and any Laurent monomial L in the weights we of N , the weights of alledges leaving v are multiplied by L, and the weights of all edges entering v aremultiplied by L−1. Clearly, the weights of paths between boundary vertices, andhence boundary measurements, are preserved under this action. Therefore, theboundary measurement map can be factorized through the space FN defined asthe quotient of EN by the action of the gauge group.

It is explained in [12] that FN can be parametrized as follows. The graph Gdivides C into a finite number of connected components called faces. The boundaryof each face consists of edges of G and, possibly, of several arcs of ∂C. A face iscalled bounded if its boundary contains only edges of G and unbounded otherwise.Given a face f , we define its face weight yf =

∏e∈∂f w

γee , where γe = 1 if the

direction of e is compatible with the counterclockwise orientation of the boundary∂f and γe = −1 otherwise. Face weights are invariant under the gauge group action.Then FN is parametrized by the collection of all face weights (subject to condition∏

f yf = 1) and a weight of an arbitrary path in G (not necessary directed) joiningtwo boundary circles; such a path is called a trail .

Below we will frequently use elementary transformations of weighted networksthat do not change the boundary measurement matrix. They were introduced byPostnikov in [32] and are presented in Fig. 4.

y1 w2

w4

w1 w2

w3 w4

w1 w2

w3

w4

w3

w4w2w111+

w2w111+1

w2w111+

w1

w2w111+

w2

w1 w2

w3 w43w

w1 w2

w4

y

Type 1

Type 2

Type 3y x

y’ x’

x

w

3w

1 1

11

1

1 1

1 1x

x

y

x y

y

1

x’

x

y’

Figure 4. Postnikov transformations


Another important transformation is path reversal : for a given closed path onecan reverse the directions of all its edges and replace each weight wi with 1/wi.Clearly, path reversal preserves face weights. The transformations of boundarymeasurements under path reversal are described in [32, 10, 12].

As was shown in [10, 12], the space of edge weights can be made into a Pois-son manifold by considering Poisson brackets that behave nicely with respect toa natural operation of concatenation of networks. Such Poisson brackets on ENform a 6-parameter family, which is pushed forward to a 2-parameter family ofPoisson brackets on FN . Here we will need a specific member of the latter family.The corresponding Poisson structure, called standard, is described in terms of thedirected dual network N ∗ defined as follows. Vertices of N ∗ are the faces of N .Edges of N ∗ correspond to the edges of N that connect either two internal verticesof different colors, or an internal vertex with a boundary vertex; note that theremight be several edges between the same pair of vertices in N ∗. An edge e∗ in N ∗

corresponding to e in N is directed in such a way that the white endpoint of e (ifit exists) lies to the left of e∗ and the black endpoint of e (if it exists) lies to theright of e. The weight w∗(e∗) equals 1 if both endpoints of e are internal vertices,and 1/2 if one of the endpoints of e is a boundary vertex. Then the restriction ofthe standard Poisson bracket on FN to the space of face weights is given by

(3.1) yf , yf ′ =

∑

e∗:f→f ′

w∗(e∗)−∑

e∗:f ′→f

w∗(e∗)

yfyf ′ .

The bracket of the trail weight z and a face weight yf is given by z, yf = cfzyf .The description of cf in the general case is rather lengthy. We will only need it inthe case when the trail is a directed path P in G. In this case

(3.2) z, yf =∑

P ′⊂P

±

∑

e∈P ′,e∗:f→f ′

w∗(e∗)−∑

e∈P ′,e∗:f ′→f

w∗(e∗)

zyf ,

where each P ′ is a maximal subpath of P that belongs to ∂f , and the sign beforethe internal sum is positive if f lies to the right of P ′ and negative otherwise.

Any network N of the kind described above gives rise to a network N on atorus. To this end, one identifies boundary circles in such a way that the endpointsof the cut are glued together, and the ith source in the clockwise direction fromthe endpoint of the cut is glued to the ith sink in the clockwise direction from theopposite endpoint of the cut. The resulting two-valent vertices are then erased,so that every pair of glued edges becomes a new edge with the weight equal tothe product of two edge-weights involved. Similarly, n pairs of unbounded facesare glued together into n new faces, whose face-weights are products of pairs offace-weights involved. We will view two networks on a torus as equivalent if theirunderlying graphs differ only by orientation of edges, but have the same vertexcoloring and the same face weights. The parameter space we associate with Nconsists of face weights and the weights zρ, z of two trails Pρ and P . The firstof them is homological to the closed curve on the torus obtained by identifyingendpoints of the cut, and the second is noncontractible and not homological to thefirst one. The standard Poisson bracket induces a Poisson bracket on face-weightsof the new network, which is again given by (3.1) with the dual graph N ∗ replacedby N ∗ defined by the same rules. The bracket between zρ or z and face-weights is


given by (3.2), provided the corresponding trails are directed paths in G. Finally,under the same restriction on the trails,

(3.3) z, zρ =∑

P ′

cP ′zzρ,

where each P ′ is a maximal common subpath of P and Pρ and cP ′ is defined inFig. 5.

PP

P PP

P

P

P

PP

P

P

a) c =0

ρ ρ

ρ c) c =1

b) c =−1

Figure 5. To the definition of cP ′

3.2. The (x,y)-dynamics. Let us define a network Nk,n on the cylinder. It has ksources, k sinks, and 4n internal vertices, of which 2n are black, and 2n are white.Nk,n is glued of n isomorphic pieces, as shown in Fig. 6.

pi

pi+1

pi+k−1

Figure 6. Local structure of the networks Nk,n and Nk,n

The pieces are glued together in such a way that the lower right edge of thei-th piece is identified with the upper left edge of the (i + 1)-th piece, providedi+1 ≤ n, and the upper right edge of the i-th piece is identified with the lower leftedge of the (i + k − 1)-st piece, provided i + k − 1 ≤ n. The network Nk,n on thetorus is obtained by dropping the latter restriction and considering cyclic labelingof pieces. The faces of Nk,n are quadrilaterals and octagons. The cut hits onlyoctagonal faces and intersects each white-white edge. The network N3,5 is shownin Fig. 7. The figure depicts a torus, represented as a flattened two-sided cylinder(the dashed lines are on the “invisible” side); the edges marked by the same symbolare glued together accordingly. The cut is shown by the thin line. The meaning ofthe weights xi and yi will be explained later.


x1

y1x2

x x4

x5

2

y3

y4

y5y

3p1

%

*

#

*

#

%

Figure 7. The network N3,5 on the torus

Proposition 3.1. The directed dual of Nk,n is isomorphic to Qk,n.

Proof. It follows from the construction above that Nk,n has 2n faces, n of themquadrilaterals and other n octagons. The quadrilateral faces correspond to p-vertices of the directed dual, and octagonal, to its q-vertices. Consider the quadri-lateral corresponding to pi. The four adjacent octagons are labelled as follows: theone to the left is qi−r−1, the one above is qi+r′ , the one to the right is qi+r′+1,and the one below is qi−r . Therefore, the octagonal face to the left of the quadri-lateral pi+1 is qi−r, and the one above it is qi+r′+1, which justifies the first gluingrule above. Similarly, the octagonal face to the left of the quadrilateral pi+k−1 isqi+k−2−r = qi+r′ , and the one below it is qi+k−1−r = qi+r′+1, which justifies thesecond gluing rule above, see Fig. 8, where the directed dual is shown with dottedlines. Therefore, we have restored the adjacency structure of Qk,n.

qi+r’

qi−r

pi+1

qi+r’+1

i+k−1p

qi−r−1 pi

Figure 8. The local structure of the directed dual of Nk,n

Corollary 3.2. The restriction of the standard Poisson bracket to the space of face

weights of Nk,n coincides with the bracket ·, ·k.

Proof. Follows immediately from (3.1) and Proposition 3.1, see Fig. 8.

Assume that the edge weights around the face pi are ai, bi, ci, and di, and allother weights are equal 1, see Fig. 9. Besides, assume that

(3.4)

n∏

i=1

bici = 1.

In what follows we will only deal with weights satisfying the above two conditions.Applying the gauge group action, we can set to 1 the weights of the upper and

the right edges of each quadrilateral face, while keeping weights of all edges with


ia i+ka

ib

ic

id i+1b

i+k−1d

Figure 9. Edge weights prior to the gauge group action

both endpoints of the same color equal to 1. For the face pi, denote by xi theweight of the left edge and by yi, the weight of the lower edge after the gauge groupaction (see Fig. 7). Put x = (x1, . . . , xn), y = (y1, . . . , yn).

Proposition 3.3. (i) The weights (x,y) are given by

(3.5) xi = aic−1i−k+1

i−1∏

j=i−k+2

b−1j c−1

j , yi = dic−1i−k+1

i∏

j=i−k+2

b−1j c−1

j .

(ii) The relation between (p,q) and (x,y) is as follows:

(3.6) pi =yixi, qi =

xi+r+1

yi+r; xi = x1

i−1∏

j=1

pjqj−r , yi = xipi.

Proof. (i) Assume that the gauge group action is given by g1i at the upper left vertexof the ith quadrilateral, by g2i at the upper right vertex, by g3i at the lower leftvertex, and by g4i at the lower right vertex. The conditions on the upper and rightedges of the quadrilateral give big

1i /g

2i = 1 and cig

2i /g

4i = 1, while the conditions

on the two external edges going right from the quadrilateral give g4i = g1i+1 and

g2i = g3i+k−1. Denote ρi = g1i /g3i . From the first three equations above we get

g3i+1 = g3i biciρi/ρi+1. Iterating this relation i+k− 1 times and taking into account

the fourth equation above we arrive at ρi+k−1 = ci∏i+k−2

j=i+1 bjcj , or

ρi = ci−k+1

i−1∏

j=i−k+2

bjcj .

Now the first relation in (3.5) is restored from xi = aig3i /g

1i = ai/ρi. To find yi

we write

yi = dig3ig4i

= dig3ig1i

g1ig2i

g2ig4i

=di

ρibici,

which justifies the second relation in (3.5). Note that n-periodicity of ρi, and henceof xi and yi, is guaranteed by condition (3.4).

(ii) The expression for pi follows immediately from the definition of face weights.Next, the face weight for the octagonal face to the right of pi is qi+r′+1 = xi+k/yi+k−1,which yields qi = xi+k−r′−1/xi+k−r′−2 = xi+r+1/xi+r. The remaining two formu-las in (3.6) are direct consequences of the first two.

Note that by (3.6), the projection πk : (x,y) 7→ (p,q) has a 1-dimensionalfiber. Indeed, multiplying x and y by the same coefficient t does not change thecorresponding p and q.

It follows immediately from (3.6) that∏n

i=1 piqi = 1, so the relevant map is T(1)

k .Let us show how it can be described via equivalent transformations of the network


Nk,n. The transformations include Postnikov’s moves of types 1, 2, and 3, and thegauge group action. We describe the sequence of these transformations below.

We start with the network Nk,n with weights xi and yi on the left and lower edgeof each quadrilateral face. First, we apply Postnikov’s type 3 move at each p-face(this corresponds to cluster τ -transformations at p-vertices of Qk,n given by (2.3)).To be able to use the type 3 move as shown in Fig. 4 we have first to conjugate itwith the gauge action at the lower right vertex, so that w1 = xi, w2 = 1/yi, w3 = 1,w4 = y. Locally, the result is shown in Fig. 10 where σi = xi + yi.

σi

x i σi−1

i+1y σi+1−1

σi+k−1

−1σi+kσi−1

y i σi−1

Figure 10. Type 3 Postnikov’s move for Nk,n

Next, we apply type 1 and type 2 Postnikov’s moves at each white-white andblack-black edge, respectively. In particular, we move vertical arrows interchangingthe right-most and the left-most position on the network in Fig. 7 using the fact thatit is drawn on the torus. These moves interchange the quadrilateral and octagonalfaces of the graph thereby swapping the variables p and q, see Fig. 11.

σi+k−1

−1σi+k

i+1y σi+1−1

x i σi−1 qi+r’+1

Figure 11. Type 1 and 2 Postnikov’s moves for Nk,n

It remains to use gauge transformations to achieve the weights as in Fig. 7. Inour situation, weights ai, bi, ci, di are as follows, see Fig. 11:

(3.7) ai =xiσi, bi = σi+k−1, ci =

1

σi+k, di =

yi+1

σi+1.

Note that condition (3.4) is satisfied. This yields the map Tk, the main characterof this paper, described in the following proposition.

Proposition 3.4. (i) The map Tk is given by

(3.8) x∗i = xi−r′−1xi+r + yi+r

xi−r′−1 + yi−r′−1, y∗i = yi−r′

xi+r+1 + yi+r+1

xi−r′ + yi−r′,

(ii) The maps Tk and T(1)

k are conjugated via πk: πk Tk = T(1)

k πk.

Proof. (i) Applying relations (3.5) to the weights (3.7) we get

x∗i+r′+1 = xiσi+k−1

σi, y∗i+r′+1 = yi+1

σi+k

σi+1,


which immediately implies (3.8).(ii) Checked straightforwardly using (2.1), (3.6), and (3.8).

Remark 3.5. Note that the map Tk commutes with the scaling action of the groupR∗: (x,y) 7→ (tx, ty), and that the orbits of this action are the fibers of theprojection πk.

Maps T2 and T3 can be further described as follows. The map T2 is a periodicversion of the discretization of the relativistic Toda lattice suggested in [41]. Itbelongs to a family of Darboux-Backlund transformations of integrable lattices ofToda type, that were put into a cluster algebras framework in [13].

Proposition 3.6. The map T3 coincides with the pentagram map.

Proof. Indeed, for k = 3, (3.8) gives

(3.9) x∗i = xi−2xi + yi

xi−2 + yi−2, y∗i = yi−1

xi+1 + yi+1

xi−1 + yi−1.

Change the variables as follows: xi 7→ Yi, yi 7→ −YiXi+1Yi+1. In the new variables,the map (3.9) is rewritten as

X∗i = Xi−1

1−Xi−2Yi−2

1−XiYi, Y ∗

i = Yi1−Xi+1Yi+1

1−Xi−1Yi−1,

which becomes formula (1.1) after the cyclic shift Xi 7→ Xi+1, Yi 7→ Yi+1. Note thatthe maps Tk, and in particular the pentagram map, commute with this shift.

Similarly to what was done in the previous section, we may consider, along with

the map Tk based on p-dynamics T k, another map, based on q-dynamics Tk; it is

natural to denote this map by T k . Its definition differs from that of Tk by the order

in which the same steps are performed. First of all, type 1 and 2 Postnikov’s movesare applied, which leads to quadrilateral faces looking like those in Fig. 10. Theweights of the left and the lower edge bounding the face labeled qi are thus equalto 1, the weight of the upper edge equals yi+r, and the weight of the right edgeequals xi+r+1. Next, the type 3 Postnikov’s move is applied, followed by the gaugegroup action.

An alternative way to describe T k is to notice that the network Nk,n can be

redrawn in a different way. Recall that the network on the torus was obtained fromthe network on the cylinder by identifying the two boundary circles so that thecut ρ becomes a closed curve. Conversely, the network on the cylinder is obtainedfrom the network on the torus by cutting the torus along a closed curve. Thiscurve intersects exactly once k monochrome edges: the black monochrome edgethat points to the face p1 and k − 1 white monochrome edges that point to thefaces p1, . . . , pk−1. Alternatively, the torus can be cut along a different closed curvethat intersects the same black monochrome edge and all the n − k + 1 remainingwhite monochrome edges. An alternative representation of N3,5 is shown in Fig 12.The cut shown in Fig 12 coincides with that in Fig. 7. We can further reversethe closed path shown with the dashed line and apply type 1 and 2 Postnikovmoves at all white-white and black-black edges. It is easy to see that the resultingnetwork is isomorphic to N4,5. In general, starting with Nk,n and applying thesame transformations one gets a network isomorphic to Nn−k+2,n, which hints thatT k and Tn−k+2 are related.


y1

x1 p1

% * #

% * # &

&

x2

y2

p2

y4

x4 p4

p5

y5

x3p3

y3

x5

Figure 12. An alternative representation of N3,5

Introduce an auxiliary map Dk given by

(3.10) x∗i =1

xi+r

i+r−1∏

j=i−r′

yjxj, y∗i =

1

xi+r+1

i+r∏

j=i−r′

yjxj.

The following analog of Proposition 2.2 explains the relation between T , T−1 andT .

Proposition 3.7. (i) The maps T−1k and T

k coincide and are given by

(3.11) x∗i = xi+r′+1xi−r + yi−r−1

xi+r′+1 + yi+r′, y∗i = yi+r′

xi−r + yi−r−1

xi+r′+1 + yi+r′.

(ii) The maps Tk and T k are almost conjugated by Dk:

(3.12) Sr−r′ Tk Dk = Dk Tk.

(iii) Let Dk,n be given by xi = yi−r−1, yi = xi−r. Then

T k = Dk,n Tn−k+2 Dn−k+2,n.

Proof. (i) The proof of (3.11) for T−1k is similar to that of Proposition 2.2(i). It is

easy to check that the maps Dk and Dk given by (3.10) and (2.4) are conjugatedvia πk: πk Dk = Dk πk. Besides, define the map Ck by

x∗i =xi−k+1 + yi−k+1

xi−k+1(xi + yi)

i−1∏

j=i−k+2

yjxj, y∗i =

xi−k+1 + yi−k+1

xi−k+1(xi + yi)

i∏

j=i−k+2

yjxj.

Similarly, πk Ck = Ck πk. Moreover, Tk = Ck Dk. Therefore, T−1k = D−1

k C−1k .


A direct computation shows that Ck is an involution, while D−1k is given by

x∗i =1

xi+r′

i+r′−1∏

j=i−r

yjxj, y∗i =

1

xi+r′+1

i+r′∏

j=i−r

yjxj,

and (3.11) for T−1k follows.

To prove (3.11) for T k one has to perform all the steps described above, similarly

to what was done in the proofs of Propositions 3.3 and 3.4.(ii) Follows immediately from (i) and the relation D2

k = Sr−r′.(iii) Checked straightforwardly taking into account (2.7). Note that transforma-

tions Dk,n and Dn−k+2,n are related to Dk,n via πn−k+2 Dn−k+2,n = Dk,n πkand πk Dk,n = Dk,n πn−k+2.

4. Poisson structure and complete integrability

The main result of this paper is complete integrability of transformations Tk, i.e.,the existence of a Tk-invariant Poisson bracket and of a maximal family of integralsin involution. The key ingredient of the proof is the result obtained in [12] onPoisson properties of the boundary measurement map defined in Section 3.1. First,we recall the definition of an R-matrix (Sklyanin) bracket, which plays a crucialrole in the modern theory of integrable systems [28, 7]. The bracket is defined onthe space of n× n rational matrix functions M(λ) = (mij(λ))

ni,j=1 and is given by

the formula

(4.1)M(λ)⊗, M(µ)

= [R(λ, µ),M(λ) ⊗M(µ)] ,

where the left-hand is understood asM(λ)⊗, M(µ)

jj′

ii′= mij(λ),mi′j′(µ) and

an R-matrix R(λ, µ) is an operator in (Rn)⊗2

depending on parameters λ, µ andsolving the classical Yang-Baxter equation. We are interested in the bracket asso-ciated with the trigonometric R-matrix (for the explicit formula for it, which wewill not need, see [28]).

4.1. Cuts, rims, and conjugate networks. Let N be a perfect network on thecylinder; recall that N stands for the perfect network on the torus obtained fromN via the gluing procedure described in Section 3.1.

Theorem 4.1. For any perfect network N on the torus, there exists a perfect net-

work N on the cylinder with sources and sinks belonging to different components

of the boundary such that N is equivalent to N , the map EN → FN is Poisson

with respect to the standard Poisson structures, and spectral invariants of the im-

age MN (λ) of the boundary measurement map depend only on FN . In particular,

spectral invariants of MN (λ) form an involutive family of functions on FN with

respect to the standard Poisson structure.

Proof. Consider a closed simple noncontractible oriented loop γ on the torus; we

call it a rim if it does not pass through vertices of N and its intersection indexwith the cut ρ equals ±1. To avoid unnecessary technicalities, we assume that γ

and all edges of N are smooth curves. Besides, we assume that each edge intersectsγ in a finite number of points and that all the intersections are transversal. Eachintersection point defines an orientation of the torus via taking the tangent vectorsto the edge and to the rim at this point and demanding that they form a right basis


of the tangent plane. We say that the rim is ideal if its intersection points with alledges define the same orientation of the torus.

Proposition 4.2. Let N be a perfect network on the torus, then there exists a rim

which becomes ideal after a finite number of path reversals in N .

Proof. Consider the universal covering π of the torus by a plane. Take an arbitraryrim γ. The preimage π−1(γ) is a disjoint union of simple curves in the plane, eachone isotopic to a line. Fix arbitrarily one such curve l0; it divides the plane intotwo regions L and R lying to the left and to the right of the curve, respectively.Let li, i ∈ N, be the connected components of π−1(γ) lying in R: l1 is the first oneto the right of l0, l2 is the next one, etc.

Let NR be the part of the network covering N that belongs to R. Each inter-

section point of an edge of N with γ gives rise to a countable number of boundary

vertices of NR lying on l0. Denote by m the number of intersection points of γ with

the edges of N . We will need the following auxiliary statement.

Lemma 4.3. Let P be a possibly infinite oriented simple path in NR that ends at

a boundary vertex and intersects lm+1. Then there exist i, j such that m + 1 ≥i > j ≥ 0 and points ti ∈ li and tj ∈ lj on P such that ti precedes tj on P and

π(ti) = π(tj).

Proof. Let us traverse P backwards starting from its endpoint, and let tm+1 bethe first point on lm+1 that is encountered during this process. Further, let ti for0 ≤ i ≤ m be the first point on li that is encountered while traversing P forwardfrom tm+1; in particular, t0 is the endpoint of P . The proof now follows from thepigeonhole principle applied to the nested intervals of P between the points tm+1

and ti.

Assume that NR contains a path P as in Lemma 4.3. Consider the interval Pij

of P between the points ti and tj described in the lemma. Clearly π(Pij) is a closednoncontractible path on the torus. If π(Pij) is a simple path, its reversal increasesby one the number of intersection points on γ that define a right basis. If π(Pij)is not simple and s is a point of selfintersection, it can be decomposed into a pathfrom π(ti) to s, a loop through s, and a path from s to π(tj). Further, the loop canbe erased, and the remaining two parts glued together, which results in a closedpath on the torus with a smaller number of selfintersection points. After a finitenumber of such steps we arrive at a simple closed path on the torus that can bereversed.

Proceeding in this way, we get a network N ′ on the torus equivalent to N such

that any path in N ′R that ends at a boundary vertex does not intersect lm+1. Note

that a path like that may still be infinite. Each such path divides R into tworegions: one of them contains lm+1, while the other one is disjoint from it. Let A

be the intersection of the regions containing lm+1 over all paths P in N ′R, and let

∂A be its boundary. Clearly, ∂A is invariant under the translations that commutewith π and take each li into itself. Therefore, π(∂A) is a simple loop on the torus,and it is homologous to γ; it is not a rim yet since it contains edges and vertices of

N ′.Each vertex v lying on π(∂A) has three incident edges. Two of them lie on π(∂A)

as well. Since a preimage t of v belongs to ∂A, the preimages of these two edges


incident to t belong to paths that end at l0. Therefore, if the third edge incident tov is pointed towards v, its preimage incident to t should belong to the complementof A, by the definition of A.

Now, to build a rim, we take a tubular ε-neighborhood of ∂A, and consider theboundary ∂A+ε of this tubular neighborhood that lies inside A. For ε small enough,the above property of the vertices lying on π(∂A) guarantees that the rim π(∂A+ε)intersects only those edges that point from these vertices into A, and hence eachintersection point defines a right basis. Therefore π(∂A+ε) is an ideal rim.

Returning to the proof of the theorem, we apply Proposition 4.2 to find the

corresponding ideal rim on the torus. Let N be the network obtained from N ′

after we cut the torus along this rim. Note that each edge of N ′ that intersectsthe rim yields several (two or more) edges in N ; the weights of these edges arechosen arbitrarily subject to the condition that their product equals the weight of

the initial edge. By Proposition 4.2, all sources of N ′ belong to one of its boundary

circles, while all sinks belong to the other boundary circle. Besides, N = N ′, and

hence N and N are equivalent. Clearly, one can choose a new cut ρ′ on the torusisotopic to ρ such that it intersects the rim only once. Consequently, after the torusis cut into a cylinder, ρ′ becomes a cut on the cylinder.

The rest of the proof relies on two facts. One is Theorem 3.13 of [12]: for

any network on a cylinder with the equal number of sources and sinks belonging to

different components of the boundary, the standard Poisson structure on the space of

edge weights induces the trigonometric R-matrix bracket on the space of boundary

measurement matrices. The second is a well-known statement in the theory ofintegrable systems: spectral invariants of MN (λ) are in involution with respect to

the Sklyanin bracket , see Theorem 12.24 in [28].

We can now apply Theorem 4.1 to the network Nk,n. Clearly, one can choosethe rim γ in such a way that the resulting network on a cylinder will be Nk,n. Notethat in this case no path reversals are needed. For example, for the network N3,5, γcan be represented by a closed curve slightly to the left of the edge marked x1 andtransversal to all horizontal edges. The resulting networkN3,5 can be seen in Fig. 7,provided we refrain from gluing together edges marked with the same symbols andregard that figure as representing a cylinder rather than a torus. Furthermore, thisnetwork on a cylinder is a concatenation of n elementary networks of the same formshown on Fig. 13 (for the cases k = 2 and k = 3).

x

2

12 3

1

2

y

1

2

1y 3

xiλ

λi

i

i

Figure 13. Elementary networks

Since elementary networks are acyclic, the corresponding boundary measurementmatrices are

Li(λ) =

(−λxi xi + yi−λ 1

)


for k = 2 and

(4.2) Li(λ) =

0 0 0 . . . xi xi + yi−λ 0 0 . . . 0 00 1 0 . . . 0 00 0 1 . . . 0 0. . . . . . . . . . . . . . . . . .0 0 0 . . . 1 1

for k ≥ 3 (negative signs are implied by the sign conventions mentioned in Sec-tion 3.1). Consequently, the boundary measurement matrix that corresponds toNk,n is

(4.3) Mk,n(λ) = L1(λ) · · ·Ln(λ).

In our construction above, the cut ρ and the rim γ are represented by non-contractible closed curves from two distinct homology classes; to get the networkNk,n on the cylinder we start from the network Nk,n on the torus and cut it alongγ so that ρ becomes a cut in Nk,n. One can interchange the roles of ρ and γ and tocut the torus along ρ, making γ a cut. This gives another perfect network N ′

k,n onthe cylinder with n sources and n sinks belonging to different components of theboundary. To this end, we first observe that ρ intersects all → edges and noother edges of Nk,n (see Fig. 13). We label the resulting intersection points along ρby numbers from 1 to n in such a way that the point seen on Fig. 13 is labeled by i(for k ≥ 3 this point belongs to the edge that connects the source 2 with the sink 1).Next, we cut the torus along ρ. Each of the newly labeled intersection points givesrise to one source and one sink in N ′

k,n. The rim γ becomes the cut ρ′ for N ′k,n.

It is convenient to view N ′k,n as a network in an annulus with sources on the outer

boundary circle and sinks on the inner boundary circle. The cut ρ′ starts at thesegment between sinks n and 1 on the inner circle and ends on the correspondingsegment on the outer circle. It is convenient to assume that in between it crossesk − 1 edges incident to the inner boundary, followed by a single • → • edge (seeFig. 14). The variable associated with the cut in N ′

k,n will be denoted by z. We

will say that N ′k,n is conjugate to Nk,n.

Proposition 4.4. Let Dx = diag(x1, . . . , xn), Dy = diag(y1, . . . , yn), and Z =

−zen1 +∑n−1

i=1 ei,i+1. The boundary measurement matrix Ak,n(z) = (aij(z))ni,j=1

for the network N ′k,n is given by

(4.4) Ak,n(z) = Z (Dx +DyZ)Zk−2 (1n − Z)

−1.

Proof. For any source i and sink j there are exactly two simple (non-self-intersecting)directed paths in N ′

k,n directed from i to j : one contains the edge of weight xi+1,and the other, the edge of weight yi+1. Every such path has a subpath in commonwith the unique oriented cycle in N ′

k,n that contains all edges of weight 1 that arenot incident to either of the boundary components. The weight of this cycle isz, which means that all boundary measurements aij(z) acquire a common factor1 − z + z2 − . . . = 1

1+z (see the sign conventions in Section 3.1). The simple di-rected path from i to j containing the edge of weight xi+1 intersects the cut onceif k − n− 1 ≤ j − i < k − 1, twice if j − i < k − n− 1, and does not intersect thecut if j − i ≥ k − 1. The simple directed path from i to j containing the edge ofweight yi+1 intersects the cut once if k − n ≤ j − i < k, twice if j − i < k − n, and


xy

y

x

y

x

y

x

x

y

1 2

3

12

3 4

5

4

5

55

2

2

4

4

3

3

1

1

Figure 14. The conjugate network N ′3,5

does not intersect the cut if j − i ≥ k. All intersections are positive. Thus, by thesign conventions,

(1 + z)aij(z) =

xi+1 + yi+1 if j − i > k − 1,xi+1 − zyi+1 if j − i = k − 1,

−z(xi+1 + yi+1) if k − n− 1 < j − i < k − 1,−z(xi+1 − zyi+1) if j − i = k − n− 1,z2(xi+1 + yi+1) if j − i < k − n− 1.

or, equivalently,

Ak,n(z) =1

1 + z

× (diag(x2, . . . , xn, x1) + diag(y2, . . . , yn, y1)Z)(Zk−1 + . . .+ Zn+k−2

).

Here we used the relation Zn = −z1n. The claim now follows from the identities

(4.5) Z diag(d1, . . . , dn)Z−1 = diag(d2, . . . , dn, d1)

and

(4.6) (1n − Z)−1 =1

1 + z

(1n + . . .+ Zn−1

).

4.2. Poisson structure. Let Mk,n and Ak,n be images of the boundary measure-ment maps from ENk,n

and EN ′

k,nrespectively. Theorem 4.1 implies that spectral

invariants of elements of Mk,n and Ak,n viewed as functions on FNk,nare in invo-

lution with respect to the standard Poisson structure. However, quantities xn, yn,


and therefore the spectral invariants of Mk,n(λ) and Ak,n(z) are only defined asfunctions on a subset F1

Nk,nspecified by the condition (3.4).

Proposition 4.5. (i) F1Nk,n

is a Poisson submanifold of FNk,nwith respect to the

standard Poisson structure. For n ≥ 2k− 1, the restriction of the standard Poisson

structure to F1Nk,n

is given by

(4.7)xi, xi+l = −xixi+l, 1 ≤ l ≤ k − 2; yi, yi+l = −yiyi+l, 1 ≤ l ≤ k − 1;

yi, xi+l = −yixi+l, 1 ≤ l ≤ k − 1; yi, xi−l = yixi−l, 0 ≤ l ≤ k − 2,

where indices are understood mod n and only non-zero brackets are listed.

(ii) The bracket (4.7) has rank 2(n− d), where d = gcd(k − 1, n). Functions

(4.8)

nd−1∏

i=0

xs+i(k−1),

nd−1∏

i=0

ys+i(k−1), s = 1, . . . , d,

are Casimir functions for (4.7).(iii) The bracket (4.7) is invariant under the map Tk.

Proof. (i) As was explained in Section 3.1, FNk,ncan be parameterized by face

coordinates pi, qi, i = 1, . . . , n, subject to∏n

i=1 piqi = 1 and weights z, zρ of twotrails that we will choose as follows. The trail that corresponds to zρ is a directedcycle Pρ that traces the • → → • part of the boundary of each quadrilateral facepi and the immediately following • → • edge of the corresponding octagonal faceqi+r′+1, see Fig. 8. After applying the gauge action to ensure that weights of allmonochrome edges are equal to 1, we see that the weight zρ is equal to

∏ni=1 bici,

where we are using notations from Section 3.2. The weight z corresponds to thedirected cycle P that consists of the → edge separating octagonal faces qn−r

and qn+1−r followed by the → • edge of the quadrilateral p1 followed by thesubpath of Pρ that closes the cycle. (For example, for N3,5 depicted in Fig. 7, Pρ

contains the → ∗ → edge followed by the → • edge labeled by x1.) SinceF1

Nk,nis cut out from FNk,n

by condition (3.4) (or, equivalently, zρ = 1), to see that

F1Nk,n

is a Poisson submanifold of FNk,n, we need to check that Poisson brackets

of zρ with z and all face weights with respect to the standard Poisson structureare zero. For the bracket zρ, z this claim follows from (3.3): there is only onemaximal common subpath of P and Pρ, and the relative position of the paths isas on Fig. 5a). For the bracket zρ, yf the claim follows from (3.2). If f is aquadrilateral face then there is only one path P ′ in the outer sum; it consists of twoedges, and the corresponding edges of the directed dual are opposite, see Fig. 8. Iff is an octagonal face then there are two paths P ′ in the outer sum. One of themconsists of three edges, of which one is monochrome; the edges of the directed dualcorresponding to the remaining two edges of P ′ are opposite, see Fig. 8. The secondone consists of a unique monochrome edge.

Next, F1Nk,n

can be parameterized by either xi, yi, i = 1, . . . , n, or by pi, qi,

i = 1, . . . , n,∏n

i=1 piqi = 1, and z. To finish the proof of statement (i), it sufficesto show that brackets (4.7) generate the same Poisson relations among pi, qi, zas the standard Poisson structure on FNk,n

. Recall that by Corollary 3.2, thePoisson brackets between pi, qi in the standard Poisson structure coincide withthose given by ·, ·k. Furthermore, it follows from (3.2) that z, qi = 0 and


z, pj = (δ1,j − δn−k+2,j) zpj. Note that due to (3.4) and gauge-invariance ofweights of directed cycles, z = x1 on F1

Nk,n. This, together with the periodicity of

Nk,n, leads to Poisson brackets xi, pj = (δi,j − δi−k+1,j)xipj .For an n-tuple (u1, ....un), let u be the column vector (log ui)

ni=1. For two n-

tuples (u1, ....un), (v1, ....vn) of functions on a Poisson manifold, we use a shorthandnotation u, vT to denote a matrix of Poisson brackets (log ui, log vj)

ni,j=1. Note

that v, uT = −u, vT T .We can then describe the Poisson brackets pi, qj, pi, pj, qi, qj, xi, pj by

p, qT = C−r−1 + Cr′+1 − C−r − Cr′ ,

p, pT = q, qT = x, qT = 0, x, pT = 1− C1−k,(4.9)

where C = e12 + · · · + en−1n + en1 = S + en1 is an n × n cyclic shift matrix andS is an upper triangular shift matrix. Similarly, formulas in (4.7) are equivalent,provided n ≥ 2k − 1, to

Ωx := x, xT =

k−2∑

i=1

(C−i − Ci

)= (1− Ck−1)

k−2∑

i=1

C−i,

Ωy := y, yT =

k−1∑

i=1

(C−i − Ci

)= (1− Ck−1)

k−1∑

i=0

C−i,(4.10)

Ωyx := y, xT =k−1∑

i=1

(C1−i − Ci

)= (1− Ck−1)

k−2∑

i=0

C−i.

We need to check that relations (4.10) imply (4.9). This follows via a straightfor-ward calculation from relations p = y− x, q = Cr (Cx− y) induced by (3.6) (onealso needs to take into account equalities r + r′ = k − 2 and CT = C−1.)

(ii) The rank of the Poisson bracket (4.7) is equal to the rank of the matrix

Ω =

(Ωx −ΩT

yx

Ωyx Ωy

).

The claim that functions (4.8) are Casimir functions follows from (4.10) and the fact

that vectors∑n

d−1

i=0 es+i(k−1), s = 1, . . . , d, form a basis of the kernel of 1− Ck−1.Let V be the complement to that kernel in Rn spanned by vectors (vi)

ni=1 such

that∑n

d−1

i=0 vs+i(k−1) = 0 for s = 1, . . . , d. Then V is invariant under C, 1− Ck−1

is invertible on V and the rank of Ω is equal to the rank of its restriction to V ⊕V .On V ⊕ V , we define

A =

(C(C − 1)−1 −(C − 1)−1

−1 1

)

and compute

AΩAT =

(0 C1−k − 1

1− Ck−1 0

).

Since AΩAT is invertible on V ⊕ V , we conclude that the rank of (4.7) is 2(n− d).(iii) Invariance of (4.7) under the map Tk can be verified by a direct calculation.


Remark 4.6. There are formulae similar to (4.7) for Tk-invariant Poisson bracketin the case n < 2k − 1 as well. Our focus on the “stable range” n ≥ 2k − 1 will bejustified by the geometric interpretation of the maps Tk in Section 5.

4.3. Conserved quantities. The ring of spectral invariants of Mk,n(λ) is gener-ated by coefficients of its characteristic polynomial

(4.11) det(In + zMk,n(λ)) =

n∑

i=1

k∑

j=1

Iij(x, y)λizj.

(Some of the coefficients Iij are identically zero.)

Proposition 4.7. Functions Iij(x, y) are invariant under the map Tk.

Proof. Recall that in Section 3.2, Tk was described via a sequence of Postnikov’smoves and gauge transformations. Furthermore, Nk,n is obtained from Nk,n bycutting the torus into a cylinder along an ideal rim γ. Note that type 3 Postnikov’smoves and gauge transformations do not affect the boundary measurement matrix.In fact, the only transformations that do change the boundary measurements aretype 1 and 2 moves interchanging vertical edges lying on different sides of γ. For anetwork on a cylinder, moving a vertical edge past γ from left to right is equivalentto cutting at the right end of the cylinder a thin cylindrical slice containing thisedge and no other vertical edges and then reattaching this slice to the cylinder onthe left. In terms of boundary measurement matrices, this operation amounts toa matrix transformation of the form M = AB 7→ M = BA under which non-zeroeigenvalues of M and M coincide. This proves the claim.

Next, we will provide a combinatorial interpretation of conserved quantities Iijin terms of the network Nk,n. This, in turn, will allow us to clarify the relationbetween boundary measurements Mk,n(λ) and Ak,n(z) in the context of the mapTk.

Let N be a perfect network on the torus with the cut ρ, and let γ be a rim. Foran arbitrary simple directed cycle C in N we define its weight w(C) as the productof the weights of the edges in C times (−1)dλ+dzλdλzdz , where dλ and dz are theintersection indices of C with ρ and γ, respectively. The weight of a collection C ofdisjoint simple cycles is defined as w(C) = (−1)|C|

∏C∈C w(C). Finally, define the

function PN (λ, z) =∑w(C), where the sum is taken over all collections of disjoint

simple cycles.

Proposition 4.8. Let N be a perfect network on the torus with no contractible

cycles, γ be an ideal rim, and M(λ) be the m ×m boundary measurement matrix

for the network on the cylinder obtained by cutting the torus along γ. Then

(4.12) det(Im + zM(λ)) =PN (λ, z)

PN (λ, 0).

Proof. First of all, note that

(4.13) det(Im + zM(λ)) = 1 +m∑

j=1

zj∑

|J|=j

∆J(M(λ)),

where ∆J (M(λ)) is the principal j×j minor ofM(λ) with the row and column setsJ . To evaluate this minor we use the formula for determinants of weighted pathmatrices obtained in [42].


It is important to note that there are two distinctions between the definitions ofthe path weights here and in [42]. First, there is no cut in [42]. This can be overcomeby modifying edge weights: if an edge of weight w intersects the cut, then its weightis changed to λw or λ−1w, depending on the orientation of the intersection; seeChapter 9.1.1 in [11] for details. Second, the sign conventions in [42] are differentfrom those described in Section 3.1: the sign of any path is positive. However, inthe absence of contractible cycles our conventions can emulate conventions of [42].To achieve that, it suffices to apply the transformation λ 7→ −λ. Indeed, after thetorus is cut along γ, the only cycles in N that survive are those with dλ = ±1. Byour sign conventions, such cycles contribute −1 to the sign of a path. The sameresult is achieved if the contribution to the sign of a path is 1, and the weight ofthe appropriate edge is multiplied (or divided) by −λ. Paths on the cylinder thatintersect the cut ρ are treated in a similar way. Finally, the rim γ is ideal, andhence the sources and the sinks lie on different boundary circles of the cylinder.Therefore,

(4.14) ∆J(M(λ)) = ∆J,m+J(W (−λ)),

where W (−λ) is the path weight matrix built by the rules of [42] based on modifiedweights of the edges.

It follows from the main theorem of [42] that

(4.15) ∆J,m+J(W ) =

∑R sgn(R)w(R)∑C(−1)|C|w(C)

where R runs over all collections of j disjoint paths connecting sources from J withthe sinks from m + J , C runs over all collections of disjoint cycles in the networkon the cylinder, and sgn(R) is the sign of the permutation πR of size j realizedby the paths from the collection R. Equation (4.15) takes into account that thereare no contractible cycles in N , and hence any cycle that survives on the cylinderintersects any path between a source and a sink. Consequently, the denominatorin (4.15) equals PN (λ, 0).

To proceed with the numerator, assume that πR can be written as the productof c cycles of lengths l1, . . . , lc subject to l1 + · · · + lc = j. Then sgn(πR) =(−1)l1−1 · · · (−1)lc−1 = (−1)j−c. It is easy to see that on the torus, the paths fromR form exactly c disjoint cycles, and that the intersection index of the ith cyclewith γ equals li. By (4.13)–(4.15), we can write

det(Im + zM(λ)) =

∑mj=0 z

j∑

|J|=j

∑C(−1)j(−1)|C|

∏C∈C w(C)

P(λ, 0),

where the inner sum is taken over all collections C that intersect γ at the prescribedset J of points. Clearly, the numerator of the above expression equals PN (λ, z),and (4.12) follows.

Corollary 4.9. One has

(4.16) det(Ik + zMk,n(λ)) = (1 + z) det(In + λAk,n(z)) = PNk,n(λ, z).

Proof. It is easy to see that the network Nk,n does not have contractible cycles.Besides, both γ and ρ are ideal rims with respect to each other. Therefore, by


Proposition 4.8,

det(Ik + zMk,n(λ)) =PNk,n

(λ, z)

PNk,n(λ, 0)

, det(In + λAk,n(z)) =PNk,n

(λ, z)

PNk,n(0, z)

.

Next, the network Nk.n is acyclic, and so PNk,n(λ, 0) = 1. Finally, the weight of

the only simple cycle in the conjugate network N ′k,n equals z, hence PNk,n

(0, z) =

1 + z and (4.16) follows.

4.4. Lax representations. Another way to see the invariance of Iij under Tkis based on a zero curvature Lax representation with a spectral parameter. Azero curvature representation for a nonlinear dynamical system is a compatibil-ity condition for an over-determined system of linear equations; this is a pow-erful method of establishing algebraic-geometric complete integrability, see, e.g.,[6]. Even more generally, the term “Lax representation” is often used for discretesystems that can be described via a re-factorization of matrix rational functionsA(z) = A1(z)A2(z) 7→ A∗(z) = A2(z)A1(z), see, e.g., [27].

Proposition 4.10. The map Tk has a k × k zero curvature representation

L∗i (λ) = Pi(λ)Li+r−1(λ)P

−1i+1(λ)

and an n× n Lax representation

Ak,n(z) = A1(z)A2(z) 7→ A∗k,n(z) = A2(z)A1(z).

Here the Lax matrices Li(λ) and Ak,n(z) are defined by (4.2) and (4.4), respectively,and L∗

i (λ) and A∗k,n(z) are their images under the transformation Tk. The auxiliary

matrix Pi(λ) is given by

Pi(λ) =

(−xi−1

σi−1− 1

λσi

1λ

− 1σi

0

)

for k = 2 and

Pi(λ) =

0 − xi

λσi− yi+1

λσi+10 . . . 0 0

0 0 xi+1

σi+1

yi+2

σi+2. . . 0 0

. . . . . . . . . . . . . . . . . . . . .0 0 0 . . .

xi+k−4

σi+k−4

yi+k−3

σi+k−30

− 1σi+k−2

0 0 . . . 0 xi+k−3

σi+k−31

1σi+k−2

1λσi+k−1

0 . . . 0 0 0

0 − 1λσi+k−1

0 . . . 0 0 0

,

for k ≥ 3, where, as before, σi = xi + yi. Finally, Aj(z) are given by

A1(z) = ZDσ(1n − Z)−1Z−r′,(4.17)

A2(z) = Zr′+2 (Dx + ZDy)D−1σ Zk−2,

where Dσ = Dx +Dy = diag(σ1, . . . , σn).

Proof. The claim can be verified by a direct calculation using equations (3.8). It isworth pointing out, however, that expressions for Pi(λ) and Aj(z) were derived byre-casting elementary network transformations that constitute Tk as matrix trans-formation. We will provide an explanation for the n × n Lax representation andleave the details for the k × k Lax representation as an instructive exercise for aninquisitive reader.


First, we rewrite equation (3.8) for Tk in terms of Dx, Dy:

D∗x =

(Z−r′−1DxD

−1σ Zr′+1

) (ZrDσZ

−r)

= Z−r′−1(DxD

−1σ

) (Zk−1DσZ

1−k)Zr′+1,

D∗y =

(Z−r′DyD

−1σ Zr′

) (Zr+1DσZ

−r−1)

= Z−r′(DxD

−1σ

) (Zk−1DσZ

1−k)Zr′ ,

which allows one to express Zr′+1A∗k,n(z)Z

−r′−1 as

Z(DxD

−1σ

(Zk−1DσZ

1−k)Z−1 + ZDyD

−1σ Z−1

(ZkDσZ

−k))Zk−1 (1n − Z)

−1.

Denote Zr′+1A∗k,n(z)Z

−r′−1 by A♯k,n(z). If we find A♯

1(z), A♯2(z) such that

Ak,n(z) = A♯1(z)A

♯2(z), A♯

k,n(z) = A♯2(z)A

♯1(z),

then A1(z) = A♯1(z)Z

−r′−1, A2(z) = A♯2(z)Z

r′+1 will provide the desired Lax rep-resentation.

Consider the transformation of the network N ′k,n induced by performing type 3

Postnikov’s move at all quadrilateral faces followed by performing type 1 Postnikov’smove at all white-white edges. The resulting network is shown in Fig. 15.

σ

σ

σ

σ

σ

1/σ

/σx

y /σ

y /σ

/σx

/σy

/σx

y /σ/σx

/σx

/σy

1 2

3

12

3 4

5

4

5

1/σ

1/σ

1/σ

1/σ 34

5

1

2

5

4

3

1

2

5

2 2

2 2

3 3

3 3

4 4

4 4

5 55 5

1 1

1 1

4

3

2

1

Figure 15. The conjugate network N ′3,5 after Postnikov’s moves

To obtain a factorization of Ak,n(z), we will view the latter network as a concate-nation of two networks glued across a closed contour that intersects all white-whiteedges and no other edges (in Fig. 15 it is represented by the dashed circle). Inter-section points are labeled 1 through n counterclockwise with the label i attached tothe point in the white-white edge incident to the edge of weight σi+1. Furthermore,


we adjusted the cut in such a way that it crosses the dashed circle through thesegment between points labeled 1 and n.

Let A♯1(z) and A♯

2(z) be boundary measurement matrices associated with the

outer and the inner networks obtained this way. Clearly, Ak,n(z) = A♯1(z)A

♯2(z).

To be able to perform the last sequence of transformation involved in Tk, namelyto apply type 3 Postnikov’s move to all black-black edges, we have to cut the torusin a different way: we first need to separate two networks along the dashed circleand then glue the outer boundary of the outer network to the inner boundary ofthe inner network matching the labels of boundary vertices. But this means that

A♯k,n(z) = A♯

2(z)A♯1(z).

It remains to check that expressions for A♯1(z), A

♯2(z) are consistent with (4.17).

Note that the cut crosses the dashed circle between the intersection points labeled

n − r′ − 1 and n − r′. The network that corresponds to A♯1(z) contains a unique

oriented cycle of weight z and, for any i, j, a unique simple directed path from sourceto sink j that contains the edge of weight σi+1. This path does not intersect the cut

if i ≤ j, otherwise it intersects the cut once. Thus, the (i, j) entry of A♯1(z) is equal

σi+1

1+z for i ≤ j and −σi+1z1+z for i > j, which means that A♯

1(z) = ZDσZ−1(1n−Z)−1

as needed.The network that corresponds to A♯

2(z) contains no oriented cycles. The sourcei is connected by a directed path of weight xi/σi to the sinks i + k − 1 and by adirected path of weight yi+1/σi+1 to the sink i+ k. The former path intersects thecut if and only if n − k + 2 ≤ i ≤ n, and the latter path intersects the cut if and

only if n − k + 1 ≤ i ≤ n. We conclude that A♯2(z) = Z (Dx + ZDy)D

−1σ Zk−2, as

needed.

In view of Proposition 4.10, the preservation of spectral invariants of Mk,n(λ)(called, in this context, the monodromy matrix ) and Ak,n(λ) is obvious. In partic-ular,

M∗k,n = P1Lr · · ·LnL1 · · ·Lr−1P

−11 .

Remark 4.11. 1. Two representations we obtained for Tk give an example of what inintegrable systems literature is called dual Lax representations. A general techniquefor constructing integrable systems possessing such representations based on dualmoment maps was developed in [1].

2. For k = 3, we obtain a 3× 3 zero-curvature representation for the pentagrammap alternative to the one given in [40].

4.5. Complete integrability.

Theorem 4.12. The map Tk is completely integrable.

Proof. Proposition 4.7 shows that spectral invariants of Mk,n(λ) (equivalently, byCorollary 4.9, spectral invariants of Ak,n(z)) are conserved quantities for Tk, whileTheorem 4.1 and Proposition 4.5 imply that conserved quantities Poisson-commute.To establish complete integrability, we need to prove that this Poisson-commutativefamily is maximal.

By Proposition 4.5, the number of Casimir functions for our Poisson structureis 2d, where d = gcd(k − 1, n). Therefore, we need to show that among spec-tral invariants of Ak,n(z) there are n + d independent functions of x = (xi)

ni=1,

y = (yi)ni=1. Furthermore, among Casimir functions described in (4.8) there are d


independent functions that depend only on y. Hence it suffices to prove that gra-dients of functions fs(x,y) =

1s+1 TrA

s+1k,n (z), s = 0, . . . , n− 1, viewed as functions

of x are linearly independent for almost all x,y or, equivalently, since fs(x,y) arepolynomials, for at least one point x,y. Using the formula for variation of tracesof powers of a square matrix

δTrAs+1 = Tr

s∑

i=0

AiδAAs−i = (s+ 1)Tr(δAAs),

we deduce that the ith component of the gradient ∇xfs(x,y) with respect to x isequal to the ith diagonal element of the matrix

Zk−1(1n − Z)−1((Dx +DyZ)Z

k−1(1n − Z)−1)s.

In particular,

(∇xfs(x,−x))i =(Zk−1(1n − Z)−1

(DxZ

k−1)s)

ii

=1

1 + z

(Zk−1(1n + . . .+ Zn−1)

(DxZ

k−1)s)

ii

=1

1 + z

(Zk−1+l

(DxZ

k−1)s)

ii

=(−z)⌈

(s+1)(k−1)n

⌉

1 + z

s∏

β=1

xi−β(k−1).

Here the second line follows from (4.6), in the third line l + (s + 1)(k − 1) =0 mod n, since only one of the powers of Z present in the second line contributesto the diagonal, and it is the one with the exponent divisible by n, the fourthline is obtained by repeated application of (4.5), and the index i − β(k − 1) isunderstood mod n. Prefactors depending on z play no role in analyzing linearindependence of ∇xfs(x,−x), so we ignore them and form an n × n matrix F =(∏s

β=1 xi−β(k−1)

)ns,i=1

. We need to show that detF is generically nonzero. We

further specialize by setting xd+1 = · · · = xn = 1, then columns of F become

Fi+α(k−1) = col(1, . . . , 1︸︷︷︸α

, xi, . . . , xi︸︷︷︸n/d

, x2i , . . . , x2i︸︷︷︸

n/d

, . . .)

for i = 1, . . . , d, α = 1, . . . , n/d. Now the standard argument (akin to the one usedin computing Vandermonde determinants) shows that up to a sign detF is equal

to∏d

i=1(xi − 1)n/d−1∏

1≤i<j≤d(xi − xj)n/d. The proof is complete.

Corollary 4.13. Any rational function of Iij that is homogeneous of degree zero

in variables x, y, depends only on p, q and is preserved by the map T k. On the level

set Πpiqi = 1, such functions generate a complete involutive family of integrals

for the map T k.

Remark 4.14. In general, these functions define a continuous integrable system on

level sets of the form Πpi = c1,Πqi = c2, and the map T(c)

k intertwines theflows of this system on different level sets lying on the same hypersurface c1c2 = c.

Numerical evidence suggests that T(c)

k is not integrable whenever c 6= 1. Indeed, theleft part of Fig. 16 demonstrates the typical integrable behavior while the right partclearly shows at least three accumulation points which contradicts integrability.


Figure 16. Integrable and non-integrable dynamics

4.6. Spectral curve. We could have deduced independence of the integrals of Tkfrom the properties of the spectral curve

(4.18)

PNk,n(λ, z) = det(Ik + zMk,n)(λ)) = (1 + z) det(In + λAk,n(z)) =

∑

i,j

Iijλizj = 0

in the spirit of [28], Section 9. We will briefly discuss this spectral curve hereto point out parallels between our approach and that of [18]. There, the startingpoint for constructing an integrable system is a centrally symmetric polygon ∆ withvertices in Z

2 which gives rise to a dimer configuration on a torus whose partition

function serves as a generating function for Casimirs and integrals in involutionand defines an algebraic curve with a Newton polygon ∆.

Proposition 4.15. The Newton polygon ∆ of the spectral curve (4.18) is a paral-

lelogram with vertices (0, 0), (0, 1), (n, k), (n, k − 1).

Proof. Obviously, (0, 0), (0, 1) ∈ ∆. Since

detAk,n(z) =(−z)k−1

1 + z

(∏

i

xi − (−1)nz∏

i

yi

)

by (4.4), (n, k), (n, k − 1) are in ∆.Let (i, j) ∈ ∆; by the construction of PNk,n

(λ, z), it corresponds to a family C

of disjoint simple cycles in Nk,n that has in total i intersection points with ρ and jintersection points with γ. Consider a simple directed cycle Ct ∈ C that intersectsthe cut ρ at points numbered α1, . . . , αit+1 = α1 (we list them in the order theyappear along Ct starting with the smallest number and denote this list A(Ct)). It isconvenient to visualize Ct using the network N ′

k,n. Then one can see that αs+1 can

be expressed as αs+1 = αs+βs mod n, where s = 1, . . . , it and βs ∈ [k−1, n+k−2],see Fig. 14. We thus have it(k− 1) ≤ β1 + · · ·+ βit = jtn for some jt, which meansthat Ct intersects the rim γ exactly jt times. Consequently, it and jt are subject tothe inequality k−1

n it ≤ jt. Summing up over all cycles in C and taking into account

that∑it = i,

∑jt = j, we get k−1

n i ≤ j.On the other hand, each shift βs as above prohibits at least βs − k indices from

entering the set A(C) for any cycle C ∈ C; more exactly, if βs > k then theprohibited indices are αs + k, . . . αs + βs − 1, otherwise there are no such indices.So, totally at least jtn− itk indices are prohibited. Clearly, the sets of prohibited


indices for distinct cycles are disjoint. Therefore, altogether at least jn− ik indicesare prohibited and i indices are used, hence jn− i(k − 1) ≤ n. Thus, if (i, j) ∈ ∆,then k−1

n i ≤ j ≤ k−1n i + 1, which, together with an obvious inequality 0 ≤ i ≤ n,

proves the claim.

There are 2(d+ 1) integer points on the boundary of ∆:(ln

d, lk − 1

d

),

(ln

d, lk − 1

d+ 1

), l = 0, . . . , d.

The number of interior integer points (equal to the genus of the spectral curve) isn − d. The coefficient of PNk,n

(λ, z) that corresponds to a point of the first typeis a sum where each term is the product of weights of l disjoint cycles, each ofthem characterized uniquely by i = n/d, α1 ∈ [1, d], and β1 = . . . = βn/d = k − 1

(see Fig. 14). The weight of such cycle is the Casimir function∏n/d−1

s=0 xα1+s(k−1),cp. with the first expression in (4.8).

On the other hand, any term contributing to the coefficient corresponding toa point of the second type is the weight of a collection of disjoint cycles that isrepresented in N ′

k,n by a unique collection of non-intersecting paths joining ln/d

sources α1, α2, . . . , αl, α1 + (k − 1), α2 + (k − 1), . . . , αl + (k − 1), . . . to sinks α2 +(k − 1), α3 + (k − 1), . . . , αl + (k − 1), α1 + 2(k − 1), . . ., respectively, where 1 ≤α1 < . . . < αl ≤ d. The weight in question is the product of Casimir functions∏n/d−1

s=0 yαt+s(k−1) for t = 1, . . . , l, cp. with the second expression in (4.8).Thus, just like in [18], interior points of ∆ correspond to independent integrals

while integer points on the boundary of ∆ correspond to Casimir functions.

5. Geometric interpretation

In this section we give a geometric interpretations of the maps Tk. The casesk ≥ 3 and k = 2 are different and are treated separately.

5.1. The case k ≥ 3.

5.1.1. Corrugated polygons and generalized higher pentagram maps. As we alreadymentioned, a twisted n-gon in a projective space is a sequence of points V = (Vi)such that Vi+n = M(Vi) for all i ∈ Z and some fixed projective transformation Mcalled the monodromy. The projective group naturally acts on the space of twistedn-gons. Let Pk,n be the space of projective equivalence classes of generic twisted

n-gons in RPk−1, where “generic” means that every k consecutive vertices do notlie in a projective subspace. Clearly, the space Pk,n has dimension n(k − 1).

We say that a twisted polygon V is corrugated if, for every i, the vertices Vi,Vi+1, Vi+k−1 and Vi+k span a projective plane. The projective group preserves thespace of corrugated polygons. Projective equivalence classes of corrugated polygonsconstitute an algebraic subvariety of the moduli space of polygons in the projectivespace. Note that a polygon in RP2 is automatically corrugated.

Denote by P0k,n ⊂ Pk,n the space of projective equivalence classes of corrugated

polygons satisfying the additional genericity assumption that, for every i, everythree out of the four vertices Vi, Vi+1, Vi+k−1 and Vi+k are not collinear.

Lemma 5.1. One has: dim P0k,n = 2n.


Proof. As it was already mentioned, dim Pk,n = n(k − 1). For each i = 1, . . . , n,one has a constraint: the vertex Vi+k lies in the projective plane spanned by Vi,

Vi+1, Vi+k−1. The codimension of a plane in RPk−1 is k − 3, which yields k − 3equations. Thus

dim P0k,n = n(k − 1)− n(k − 3) = 2n,

as claimed.

The consecutive (k− 1)-diagonals (the diagonals connecting Vi and Vi+k−1) of acorrugated polygon intersect, and the intersection points form the vertices of a newtwisted polygon: the ith vertex of this new polygon is the intersection of diagonals(Vi, Vi+k−1) and (Vi+1, Vi+k). This (k− 1)-diagonal map commutes with projectivetransformations, and hence one obtains a rational map Fk : P0

k,n → Pk,n. (Note

that this rational map is well defined only on an open subset of P0k,n because the

image polygon may be degenerate.) F3 is the pentagram map; the maps Fk fork > 3 are called generalized higher pentagram maps.

Remark 5.2. Corrugated polygons for k > 2 were independently defined by M.Glick(private communication).

Given a corrugated polygon V , one can also construct a new polygon whose ithvertex is the intersection of the lines (Vi, Vi+1) and (Vi+k−1, Vi+k). This defines amap Gk : P0

k,n → Pk,n.Similarly to above, one can define spaces of twisted and corrugated polygons in

the dual projective space (RPk−1)∗, as well as dual analogs of the maps Fk andGk; in what follows, the objects in the dual space will be marked by an asterisk.Besides, we will need the notion of the projectively dual polygon. Let V be ageneric polygon in RPk−1. Each consecutive (k − 1)-tuple of vertices spans a

projective hyperplane, that is, a point of (RPk−1)∗. This ordered collection ofpoints represents the vertices of the dual polygon W = V ∗; more exactly, theprojective hyperplane spanned by Vi, . . . , Vi+k−2 represents the vertex Wi. Wedenote the projective duality map that takes V to W by ∆k.

Proposition 5.3. (i) The image of a corrugated polygon under Fk and under Gk

is a corrugated polygon.

(ii) Up to a shift of indices by k, the maps Fk and Gk are inverse to each other.

(iii) The polygon projectively dual to a corrugated polygon is corrugated.

(iv) Projective duality ∆k conjugates the maps Fk and G∗k.

Proof. (i) Let V ′i denote the ith vertex of Fk(V ). We claim that the vertices V ′

i ,V ′i+1, V

′i+k−1, V

′i+k lie in a projective plane. Indeed, V ′

i and V ′i+1 belong to the line

(Vi+1, Vi+k), and V′i+k−1 and V ′

i+k to the line (Vi+2k−1, Vi+k). These lines interestat point Vi+k, hence the the points V

′i , V

′i+1, V

′i+k−1, V

′i+k are coplanar, see Fig. 17.

The argument for the map Gk is analogous.(ii) Follows immediately from the definition of Fk and Gk, see Fig. 17.

(iii) Lift points in RPk−1 to vectors in Rk; the lift is not unique and is definedup to a multiplicative factor. We use tilde to indicate a lift of a point. A lift of a

twisted polygon is also twisted: Vi+n = M(Vi) for all i, where M ∈ GL(k,R) is alift of the monodromy.


V

V

V

V V

V

i

Vi+1

Vi+2i+k

i+k+1

i+2k−2

i+2k−1i+2k

i+1i

i+k

V

V

i+k−1V

i+k−1V

V

Figure 17. The maps Fk and Gk

Fix a volume form in Rk. Then (k − 1)-vectors are identified with covectors. In

particular, if Vi, . . . , Vi+k−2 are points in RPk−1 spanning a hyperplane then the

respective point Wi of the dual space (RPk−1)∗ lifts to Wi = Vi ∧ · · · ∧ Vi+k−2.Let V be a generic corrugated polygon. We need to prove that the (k−1)-vectors

Wi, Wi+1, Wi+k−1 and Wi+k are linearly dependent for all i.

Since the polygon V is corrugated, Vi+2k−2 ∈ span(Vi+2k−3, Vi+k−2, Vi+k−1),and hence

Wi+k = Vi+k ∧ · · · ∧ Vi+2k−2 ∈ span(Vi+k−2 ∧ Vi+k ∧ · · · ∧ Vi+2k−3, Wi+k−1).

In its turn,

Vi+k−2 ∧ Vi+k ∧ · · · ∧ Vi+2k−3 = Vi+k−3 ∧ Vi+k−2 ∧ Vi+k ∧ · · · ∧ Vi+2k−4 =

· · · = Vi+1 ∧ · · · ∧ Vi+k−2 ∧ Vi+k ∈ span(Vi ∧ · · · ∧ Vi+k−2, Vi+1 ∧ · · · ∧ Vi+k−1).

The latter two (k − 1)-vectors are Wi and Wi+1, as needed.(iv) We need to prove that G∗

k ∆k = ∆k Fk.

As before, we argue about lifted vectors. Let V be a lifted polygon, and letW be the polygon dual to V . The vertices of a lift of the polygon W ′′ = G∗

k(W )

are represented by vectors in span(Wi, Wi+1) ∩ span(Wi+k−1, Wi+k). Let W′′i be a

vector spanning this line. Then

W ′′i = αWi + βWi+1 = λWi+k−1 + µWi+k

for some coefficients α, β, λ, µ. Using the definition of the points Wi, we have:

W ′′i = (αVi ± βVi+k−1) ∧ Vi+1 ∧ . . . ∧ Vi+k−2

= (λVi+k−1 ± µVi+2k−2) ∧ Vi+k ∧ . . . ∧ Vi+2k−3.(5.1)

On the other hand, the vertex V ′i of Fk(V ) is the intersection point of the lines

(Vi, Vi+k−1) and (Vi+1, Vi+k). Let V′i be a lift of this point; then V ′

i = sVi+1+ tVi+k

where s and t are coefficients. We want to show that ∆k(V′i ) =W ′′

i , that is, using

the identification of (k − 1)-vectors with covectors, that V ′i ∧ W ′′

i = 0. Indeed, inview of (5.1),

V ′i ∧ W ′′

i = sVi+1 ∧ (αVi ± βVi+k−1) ∧ Vi+1 ∧ . . . ∧ Vi+k−2+

tVi+k ∧ (λVi+k−1 ± µVi+2k−2) ∧ Vi+k ∧ . . . ∧ Vi+2k−3 = 0,


as claimed.

Remark 5.4. Let us briefly mention a natural continuous analog of corrugated poly-gons and the generalized higher pentagram maps. A twisted curve in RPk−1 is amap γ : R → RPk−1 such that γ(x + 1) = M(γ(x)) for all x where M is a fixedprojective transformation. The projective group naturally acts on twisted curves,and one considers the projective equivalence classes.

A curve is called c-corrugated if the tangent lines at points γ(x) and γ(x+ c) arecoplanar (not skew) for all x ∈ R. We claim that the line connecting points γ(x)and γ(x+ c) envelops a new curve, γ(x). Indeed, the coplanarity condition impliesthat γ(x+ c)− γ(x) = u(x)γ′(x) + v(x)γ′(x+ c) for some functions u and v. Thenthe envelope γ is given by the equation

γ(x) =u(x)

u(x) + v(x)γ(x) +

v(x)

u(x) + v(x)γ(x+ c),

as can be easily verified by differentiation.Thus we obtain a map Fc : γ 7→ γ. It is even easier to describe a map Gc defined

by a c-corrugated curve γ: it is traced by the intersection points of the tangent linesat points γ(x) and γ(x+ c). These maps commute with projective transformations.Of course, the notion of corrugated curve is interesting only when k − 1 ≥ 3.

An analog of Proposition 5.3 holds; in particular, Fc(γ) is again a c-corrugatedcurve. The dynamics of the projective equivalence classes of corrugated curvesunder the transformations Fc and Gc is an interesting subject; we do not dwell onit here.

5.1.2. Coordinates in the space of corrugated polygons. Now we introduce coordi-nates in P0

k,n.

Proposition 5.5. One can lift the vertices of a generic corrugated polygon V so

that, for all i, one has:

(5.2) Vi+k = yi−1Vi + xiVi+1 + Vi+k−1,

where xi and yi are n-periodic sequences. Conversely, n-periodic sequences xi andyi uniquely determine the projective equivalence class of a twisted corrugated n-gonin RPk−1.

Proof. Consider a lifted twisted polygon V . Since V is corrugated, one has

(5.3) Vi+k = ai+k−1Vi+k−1 + bi+1Vi+1 + ciVi

for all i. The sequences ai, bi and ci are n-periodic and, due to the genericityassumption, none of these coefficients vanish.

We wish to choose the lift in such a way that the coefficient a identically equals 1.

Rescale: Vi = λiVi, where λi 6= 0. Then for (5.3) to become (5.2), the followingrecurrence should hold for the scaling factors: λi+1 = aiλi. Set λ0 = 1 anddetermine λi, i ∈ Z, by the recurrence.

After this rescaling, the coefficients change as follows:

ci 7→ci

aiai+1 · · · ai+k−1, bi+1 7→

bi+1

ai+1 · · ·ai+k−1.

Hence

(5.4) xi =bi+1

ai+1 · · · ai+k−1, yi =

ci+1

ai+1 · · · ai+k.


Thus we obtain recurrence (5.2) with n-periodic coefficients uniquely determinedby the projective equivalence class of the twisted corrugated polygon.

Conversely, given n-periodic sequences xi and yi, choose a frame V0, . . . , Vk−1 in

Rk and use recurrence (5.2) to construct a bi-infinite sequence of vectors Vi. The

periodicity of the sequences xi and yi implies that the polygon V is twisted, andrelation (5.2) implies that it is corrugated. A different choice of a frame results

in a linearly equivalent polygon and, after projection to RPk−1, in a projectivelyequivalent polygon V .

The next theorem interprets the map Tk as the generalized higher pentagrammap Fk.

Theorem 5.6. (i) In the (x,y)-coordinates, the maps Fk and Gk coincide with Tkand T−1

k up to a shift of indices. More exactly, if St : P0k,n → P0

k,n is the shift by t

in the positive direction, then Fk = Tk Sr′+1 and Gk = T−1k Sr+1.

(ii) In the (x,y)-coordinates, the projective duality ∆k coincides with Dk up to

a sign and a shift of indices. More exactly, ∆k = (−1)kDk Sr′ .

Proof. (i) Let V be a polygon in Rk satisfying (5.2). Then

Vj+k−1 + yj−1Vj = Vj+k − xj Vj+1

for j = i, i+1, i+k−1, i+k. It follows that, as lifts of points V ′i , V

′i+1, V

′i+k−1, V

′i+k

in Fig. 17, one may set:

(5.5)V ′i = Vi+k − xiVi+1, V ′

i+1 = Vi+k + yiVi+1,

V ′i+k−1 = Vi+2k−1 − xi+k−1Vi+k, V ′

i+k = Vi+2k−1 + yi+k−1Vi+k.

One has a linear relation

V ′i+k = aV ′

i+k−1 + bV ′i+1 + cV ′

i .

Substitute from (5.5) to obtain

Vi+2k−1+yi+k−1Vi+k = a(Vi+2k−1−xi+k−1Vi+k)+b(Vi+k+yiVi+1)+c(Vi+k−xiVi+1)

and use linear independence of the vectors Vi+1, Vi+k, Vi+2k−1 to conclude that

a = 1, b = xixi+k−1 + yi+k−1

xi + yi, c = yi

xi+k−1 + yi+k−1

xi + yi.

Thus the vectors V ′i , V

′i+1, V

′i+k−1, V

′i+k satisfy recurrence (5.2) with the coefficients

x′i = xixi+k−1 + yi+k−1

xi + yi, y′i = yi+1

xi+k + yi+k

xi+1 + yi+1.

This differs from (3.8) only by shifting indices by r′ + 1.The statement about Gk follows immediately from Fk = Tk Sr′+1 and Propo-

sition 5.3(ii).

(ii) We use the same notation as in Proposition 5.3. Let V be a polygon in Rk

satisfying (5.2), and W be a lift of the dual polygon.Going by the proof of Proposition 5.3(iii), line by line, we obtain the equalities:

Wi+k = (−1)kyi+k−3Vi+k−2 ∧ Vi+k ∧ · · · ∧ Vi+2k−3 + (−1)kxi+k−2Wi+k−1,

and

Vi+k−2 ∧ Vi+k ∧ · · · ∧ Vi+2k−3 = (−1)kyi−1 · · · yi+k−4Wi + yi · · · yi+k−4Wi+1.


Hence

Wi+k = (−1)kxi+k−2Wi+k−1 + (−1)kyi · · · yi+k−3Wi+1 + yi+1 · · · yi+k−3Wi.

Using formulas (5.4), we find

x∗i = (−1)kyi · · · yi+k−3

xi · · ·xi+k−2, y∗i = (−1)k

yi · · · yi+k−2

xi · · ·xi+k−1,

which differs from (3.10) only by the sign and the shift of indices by r′.

Remark 5.7. 1. In view of Theorem 5.6 and Proposition 3.7(i,ii), we may identifythe maps T

k and G∗k.

2. It would be interesting to provide a geometric interpretation for Proposi-tion 3.7(iii), which connects the maps G∗

k and Fn−k+2.

Statement (i) of Theorem 5.6, along with Theorem 4.12, implies that the gener-alized higher pentagram map Fk is completely integrable.

Integrals of the pentagram map (1.1) were constructed by R. Schwartz in [36].He observed that the map commutes with the scaling

Xi 7→ λXi, Yi 7→ λ−1Yi,

and that the conjugacy class of the monodromy of a twisted polygon is invariantunder the map. Decomposing the characteristic polynomial of the monodromy intothe homogeneous components with respect to the scaling, yields the integrals.

The next proposition shows that the integrals of the generalized higher penta-gram map can be constructed in a similar way.

Proposition 5.8. The integrals of Fk can be obtained from the monodromy M as

the homogeneous components of its characteristic polynomial.

Proof. By Theorem 5.6 and (4.11), it suffices to check that the matrixMk,n(λ) givenby (4.3) is conjugate-transpose to the scaled monodromy M(λ) of the respectivetwisted corrugated polygon.

Let us start with the monodromy. We use vectors V1, . . . , Vk as a basis in thevector space Rk, and express the next vectors using (5.2). When we come to

Vn+1, . . . , Vn+k, we get the monodromy matrix as a function of the coordinatesx,y. This process is represented by the product of matrices Qi(λ) that encode one

step (Vi . . . , Vi+k−1) 7→ (Vi+1 . . . , Vi+k). By (5.2)

Qi(λ) =

0 1 0 . . . 0 00 0 1 . . . 0 0. . . . . . . . . . . . . . . . . .0 0 0 . . . 0 1

λyi−1 λxi 0 . . . 0 1

,

where λ is the scaling factor, and M(λ) = Qn(λ)Qn−1(λ) · · ·Q1(λ).Now consider Mk,n(λ)

T : by (4.3), it is the product of Li(λ)T , i = 1, . . . , n, in

the reverse order, with Li(λ) given by (4.2). Let Ai(λ) be an auxiliary matrix

Ai(λ) =

λ−1 0 0 . . . 0 00 1 0 . . . 0 00 0 1 . . . 0 0. . . . . . . . . . . . . . . . . .xi 0 0 . . . 0 1

,


then Li(−λ)T = Ai+1(λ)−1Qi+1(λ)Ai(λ), and hence

Mk,n(−λ)T = An+1(λ)

−1Qn+1(λ)qn(λ) · · ·Q2(λ)A1(λ).

Taking into account that sequences Qi(λ) and Ai(λ) are n-periodic by Proposi-tion 5.5, we get

Mk,n(−λ)T = A1(λ)

−1Q1(λ)M(λ)Q1(λ)−1A1(λ),

and the claim follows.

We complete the discussion of coordinates in the space of corrugated polygonsby showing that the coordinates (pi, qi) introduced in Section 2 can be interpretedas cross-ratios of quadruples of collinear points. We use the following definition ofcross-ratio (out of six possibilities):

(5.6) [a, b, c, d] =(a− b)(c− d)

(a− d)(b − c).

To a twisted corrugated n-gon V we assign two n-periodic sequences of cross-ratios. Consider Fig. 17 and Fig. 18 (depicting the map Gk) and observe quadruplesof collinear points. Their cross-ratios are related to the coordinates (pi, qi) asfollows.

V

ViVi+1P

Q

i

i

V

Vi+2−k

i+1−ki+kV

i+k−1

Figure 18. Cross-ratios qi

Proposition 5.9. One has:

pi = [Vi+1, V′i , Vi+k, V

′i+1], qi−r−1 = [Pi, Vi+1, Qi, Vi]

subject to∏n

i=1 piqi = 1.

Proof. Consider the lifted polygons. We saw in the proof of Theorem 5.6 that

V ′i = Vi+k − xiVi+1, V ′

i+1 = Vi+k + yiVi+1.

Choose the basis in the 2-plane so that Vi+1 = (1, 0), Vi+k = (0, 1). Then V ′i =

(−xi, 1), V ′i+1 = (yi, 1), and

[Vi+1, V′i , V

′i+1, Vi+k] = [∞,−xi, 0, yi] =

yixi

= pi,

the last equality due to Proposition 3.3(ii).Likewise, one has

Pi = Vi+1 − Vi, Qi = xiVi+1 + yi−1Vi


in Fig. 18, and this yields

[Pi, Vi+1, Qi, Vi] =xiyi−1

= qi−r−1,

the last equality again due to Proposition 3.3(ii).The condition on the product of all coordinates follows immediately.

5.1.3. Higher pentagram maps on plane polygons. One also has the skip (k − 2)-diagonal map on twisted polygons in the projective plane. Assume that the poly-gons are generic in the following sense: for every i, no three out of the four verticesVi, Vi+1, Vi+k−1 and Vi+k are collinear. The skip (k − 2)-diagonal map assigns toa twisted n-gon V the twisted n-gon whose consecutive vertices are the intersec-tion points of the lines (Vi, Vi+k−1) and (Vi+1, Vi+k). We call these maps higher

pentagram maps and denote them by Fk. Assume that the ground field is C.

Arguing as in the proof of Proposition 5.5, we lift the points Vi to vectors Vi ∈ C3

so that (5.2) holds. This provides a rational map ψ from the moduli space of twistedpolygons in CP2 to the (x,y)-space, that is, to the moduli space of corrugated

twisted polygons in CPk−1. On the latter space, the generalized higher pentagrammap Fk acts. The relation between these maps is as follows.

Proposition 5.10. (i) The map ψ is(k3

)-to-one.

(ii) The map ψ conjugates Fk and Fk, that is, ψ Fk = Fk ψ.

Proof. Given periodic sequences xi, yi, we wish to reconstruct a twisted n-gon in

C3, up to a linear equivalence. To this end, let the first three vectors V1, V2, V3form a basis, and choose V4, . . . , Vk arbitrarily, so far. Then Vk+1, and all the next

vectors, are determined by the recurrence (5.2). The monodromy M is determinedby the condition that

(5.7) M(V1) = Vn+1, M(V2) = Vn+2, M(V3) = Vn+3.

The twist condition is that

(5.8) M(Vj) = Vn+j , j = 4, . . . , k.

If this holds, then M(Vi) = Vn+i for all i. Note that (5.8) gives 3(k − 3) equations

on that many variables (the unknown vectors being V4, . . . , Vk). We shall see thatthese are quadratic equations and proceed to solving them.

The recurrence (5.2) implies that Vq =∑k

i=1 Fiq Vi where F

iq is a function of x,y.

One has: Vj = v1j V1+v2j V2+v

3j V3 where v1j , v

2j , v

3j are the components of the vector

Vj in the basis V1, V2, V3. Rewrite (5.8), using (5.7), as

v1j

(k∑

i=1

F in+1Vi

)+ v2j

(k∑

i=1

F in+2Vi

)+ v3j

(k∑

i=1

F in+3Vi

)=

k∑

i=1

F in+j Vi,

or

A(Vj) +

k∑

i=4

(F i · Vj)Vi = Cj +

k∑

i=4

gijVi, j = 4, . . . , k,


where A = (F βn+α), α, β = 1, 2, 3, is a 3 × 3 matrix, F i = (F i

n+1, Fin+2, F

in+3),

Cj = (F 1n+j , F

2n+j , F

3n+j), and g

ij = F i

n+j . Rewrite, once again, as

(5.9) A(Vj)− Cj =

k∑

i=4

[gij − (F i · Vj)] Vi.

Let B be the k × k matrix that has A in the upper left corner, the vectors−C4,−C5, . . . ,−Ck to the right of A, the vectors −(F4)

t, . . . ,−(Fk)t below A,

and the (k − 3) × (k − 3) matrix G = gij in the bottom right corner. The en-tries of B are functions of x,y. Let ξj , j = 4, . . . , k, be the k-dimensional vec-tor (V 1

j , V2j , V

3j , 0, . . . , 0, 1, 0, . . . , 0) with 1 at j-th position. Let V be the span of

ξj , j = 4, . . . , k.

Claim: the system (5.9) is equivalent to the condition that V is a B-invariantsubspace, that is, a fixed point of the action of B on the Grassmannian Gr(k−3, k).

Indeed, one has:

(5.10) B(ξj) = (A(Vj)− Cj , g4j − (F 4 · Vj), . . . , g

kj − (F k · Vj)).

If (5.9) holds then

B(ξj) =

(k∑

i=4

[gij − (F i · Vj)] Vi, g4j − (F 4 · Vj), . . . , g

kj − (F k · Vj)

)

=

k∑

i=4

[gij − (F i · Vj)] ξi,

so V is B-invariant.Conversely, if V is B-invariant then B(ξj) =

∑ki=4 αiξi with some coefficients αi.

It follows from (5.10) that αi = gij − (F i · Vj), and that

A(Vj)− Cj =

k∑

i=4

[gij − (F i · Vj)]Vi,

which is equation (5.9). This proves the claim.A generic linear transformation B has a simple spectrum and k one-dimensional

eigenspaces. One has(k3

)invariant (k − 3)-dimensional subspaces that can be

parameterized as span(ξ4, . . . , ξk). Thus one has(k3

)choices of vectors V4, . . . , Vk

for given coordinates xi, yi. In other words, the mapping ψ from the moduli spaceof twisted n-gons Pn to the x,y-space is

(k3

)-to-one. That this map conjugates the

skip (k − 2)-diagonal map Fk with the map Fk is obvious, and we are done.

It follows that if I is an integral of the map Fk then I ψ is an integral of themap Fk. Thus the integrals (4.11) provide integrals of the higher pentagram map.

5.2. The case k = 2: leapfrog map and circle pattern.

5.2.1. Space of pairs of twisted n-gons in RP1. Let Sn be the space whose pointsare pairs of twisted n-gons (S−, S) in RP1 with the same monodromy. Here S isa sequence of points Si ∈ RP1, and likewise for S−. One has: dim Sn = 2n + 3.


The group PGL(2,R) acts on Sn. Let ϕ be the map from Sn to the (x,y)-spacegiven by the formulas:(5.11)

xi =(Si+1 − S−

i+2)(S−i − S−

i+1)

(S−i − Si+1)(S

−i+1 − S−

i+2), yi =

(S−i+1 − Si+1)(S

−i+2 − Si+2)(S

−i − S−

i+1)

(S−i+1 − Si+2)(S

−i − Si+1)(S

−i+1 − S−

i+2).

Recall that we use the cross-ratio defined by formula (5.6).

Proposition 5.11. (i) The composition of ϕ with the projection π is given by the

formulas

pi = [S−i+1, Si+1, S

−i+2, Si+2], qi =

[S−i , Si+1, Si+2, S

−i+3][S

−i+1, S

−i+2, Si+2, S

−i+3]

[S−i , S

−i+1, S

−i+2, S

−i+3][S

−i+1, Si+1, Si+2, S

−i+3]

.

(ii) The image of the map π ϕ belongs to the hypersurface∏n

i=1 piqi = 1.(iii) The fibers of this maps are the PGL(2,R)-orbits, and hence the (x,y)-space

is identified with the moduli space Sn/PGL(2,R).

Proof. (i) From Proposition 3.3, we have:

(5.12) pi =yixi, qi =

xi+1

yi.

Then a direct computation using formulas (5.11) for x and y yields the result.(ii) The sequences xi and yi are n-periodic. Multiplying pi and qi from (5.12),

i = 1, . . . , n, the numerators and denominators cancel out, and the result follows.(iii) Put the sequences of points S−, S in the interlacing order:

. . . , S−i , Si, S

−i+1, Si+1, S

−i+2, . . .

and consider the cross-ratios of the consecutive quadruples:

pi−1 = [S−i , Si, S

−i+1, Si+1], ri = [Si, S

−i+1, Si+1, S

−i+2];

the first equality was proved in (i), and the second is the definition of r. The se-quences pi and ri are n-periodic and they determine the projective equivalence classof the pair (S−, S). Thus we have a coordinate system (p, r) on Sn/PGL(2,R).

We wish to show that (x,y) is another coordinate system. Indeed, one canexpress (x,y) in terms of (p, r) and vice versa:

xi =ri(1 + pi−1)

1 + ri, yi =

ripi(1 + pi−1)

1 + ri; pi =

yixi, ri =

xi−1xixi−1(1− xi) + yi−1

(we omit this straightforward computation).

5.2.2. Leapfrog transformation. Define a transformation Φ of the space Sn, actingas Φ(S−, S) = (S, S+), where S+ is given by the following local “leapfrog” rule:given a quadruple of points Si−1, S

−i , Si, Si+1, the point S

+i is the result of applying

to S−i the unique projective involution that fixes Si and interchanges Si−1 and Si+1.

Clearly, Φ commutes with projective transformations.The transformation Φ can be defined this way over any ground field, however in

RP1 we can interpret the point S+i as the reflection of S−

i in Si in the projectivemetric on the segment [Si−1, Si+1], whence the name; see Fig. 19. Recall that theprojective metric on a segment is the Riemannian metric whose isometries are theprojective transformations preserving the segment, see [5, Chap. 4]. That is, the


projective distance between points P and Q on a segment AB is as given by theformula

d(P,Q) =1

2ln

(Q −A)(B − P )

(P −A)(B −Q)

(this formula defines distance in the Cayley-Klein, or projective, model of hyperbolicgeometry; the factor 1/2 is needed for the curvature to be −1).

Si−1 SiS Si Si+1i+ −

Figure 19. Evolution of points in the projective line

Theorem 5.12. (i) The map Φ is given by the following equivalent equations:

(5.13)1

S+i − Si

+1

S−i − Si

=1

Si+1 − Si+

1

Si−1 − Si,

(5.14)(S+

i − Si+1)(Si − S−i )(Si − Si−1)

(S+i − Si)(Si+1 − Si)(S

−i − Si−1)

= −1,

(5.15)(S+

i − Si−1)(Si − S−i )(Si+1 − Si)

(S+i − Si)(Si − Si−1)(S

−i − Si+1)

= −1.

(ii) The map induced by Φ on the moduli space Sn/PGL(2,R) is the map T2 given

in (3.8).

Proof. (i) A fractional-linear involution x 7→ y with a fixed point a is given by theformula

1

x− a+

1

y − a= b

where b is some constant. Since the leapfrog involution has Si as a fixed point andswaps S−

i with S+i and Si−1 with Si+1, one has

1

S−i − Si

+1

S+i − Si

= b =1

Si−1 − Si+

1

Si+1 − Si,

which implies (5.13). That equalities (5.14) and (5.15) are equivalent to (5.13) isverified by a straightforward computation.

(ii) We need to check that

(5.16) x∗i = xi−1xi + yi

xi−1 + yi−1, y∗i = yi

xi+1 + yi+1

xi + yi,

where (x∗i , y∗i ) are given by formulas (5.11) with S− replaced by S and S by S+.

The computation is simplified by the observation that

xi + yi =(S−

i+1 − S−i )(Si+2 − Si+1)

(S−i+1 − Si+2)(S

−i − Si+1)

.

After substituting to (5.16), a direct computation reveals that the first of the equal-ities (5.16) is equivalent to the quotient of (5.14) and (5.15), whereas the second isequivalent to their product.


The leapfrog transformation can be interpreted in terms of hyperbolic geometry.Let us identify RP1 with the circle at infinity of the hyperbolic plane H2. Thenthe restrictions of hyperbolic isometries on the circle at infinity are the projectivetransformations of RP1. Accordingly, S− and S are ideal polygons in H2.

The projective transformation that interchanges the vertices Si−1 and Si+1 andfixes Si is the reflection of the hyperbolic plane in the line Li through Si, perpendic-ular to the line Si−1Si+1 (that is, the altitude of the ideal triangle Si−1SiSi+1); seeFig. 20 where we use the projective (Cayley-Klein) model of the hyperbolic plane.

Li

Si+1Si−1

Si Si−

Si+

Figure 20. Leapfrog transformation in the hyperbolic plane

The ideal polygon obtained by reflecting each vertex S−i in the respective line

Li is S+. Thus the leapfrog transformation Φ is presented as the composition of

two involutions:

(S−, S) 7→ (S+, S) 7→ (S, S+).

We note a certain similarity of the map Φ with the polygon recutting studiedby Adler [2, 3], which is also a completely integrable transformation of polygons(in the Euclidean plane or, more generally, Euclidean space). In Adler’s case, onereflects the vertex Vi in the perpendicular bisector of the diagonal Vi−1Vi+1, afterwhich one proceeds to the next vertex by increasing the index i by 1.

5.2.3. Lagrangian formulation of leapfrog transformation. The map Φ can be de-scribed as a discrete Lagrangian system. Let us recall relevant definitions, see, e.g.,[43].

Given a manifold M , a Lagrangian system is a map F : M ×M → M ×Mdefined as follows:

F (x, y) = (y, z) if and only if∂

∂y(L(x, y) + L(y, z)) = 0,

where L :M ×M → R is a function (called the Lagrangian).Many familiar discrete time dynamical systems can be described this way (for

example, the billiard ball map, for which L(x, y) = |x− y| where x and y are pointson the boundary of the billiard table).

Note that the map F does not change if the Lagrangian is changed as follows:

(5.17) L(x, y) 7→ L(x, y) + g(x)− g(y)

where g is an arbitrary function.


A Lagrangian system has an invariant pre-symplectic (that is, closed) differential2-form

ω =∑

i,j

∂2L(x, y)

∂xi∂yjdxi ∧ dyj .

The form ω does not change under the transformation (5.17).In the next proposition, assume that the n-gons in RP1 under consideration are

closed (that is, the monodromy is the identity). As before, we choose an affinecoordinate on the projective line and treat the vertices Si and S

−i as numbers. The

index i is understood cyclically, so that i = n+ 1 is the same as i = 1.

Proposition 5.13. (i) The leapfrog map Φ is a discrete Lagrangian system with

the Lagrangian

(5.18) L(S−, S) =∑

i

ln |Si − Si+1| −∑

i

ln |Si − S−i |.

(ii) The Lagrangian L changes under fractional-linear transformation

x 7→ax+ b

cx+ d

as follows: L(S−, S) 7→ L(S−, S) + g(S−)− g(S), where g(S) =∑

i ln |cSi + d|.

Proof. (i) Differentiating L(S−, S) + L(S, S+) with respect to Si yields (5.13).(ii) If S = (aS + b)/(cS + d) then

Si − Si+1 =D(Si − Si+1)

(cSi + d)(cSi+1 + d), Si − S−

i =D(Si − S−

i )

(cSi + d)(cS−i + d)

with D = ad− bc. It follows that

∑

i

(ln |Si − Si+1| − ln |Si − S−i |) =

∑

i

(ln |Si − Si+1| − ln |Si − S−i )|+

∑

i

(ln |cS−i + d| − ln |cSi + d)|,

as claimed.

Corollary 5.14. The 2-form

ω =

n∑

i=1

dS−i ∧ dSi

(S−i − Si)2

.

is a closed PSL(2,R)-invariant differential form invariant under the map Φ.

We note that the form ω is not basic, that is, it does not descend on the quotientspace Sn/PGL(2,R).

Remark 5.15. Numerical simulations show that Φ does not have integrable behavior.Figure 21 shows chaotic behavior of two quantities: the horizontal axis is S2 − S1,the vertical axis is S3 − S1.


Figure 21. Non-integrable behavior of the map Φ for n = 6

+

Si

Si+1

Si−1

Si

S−i

Figure 22. Pairwise tangent circles

5.2.4. Circle pattern. If the ground field isC then the mapping Φ can be interpretedas a certain circle pattern.

Consider Fig. 22. This figure depicts a local rule of constructing point S+i from

points Si−1, Si, Si+1 and S−i . Namely, draw the circle through points Si−1, S

−i , Si,

and then draw the circle through points Si, Si+1, tangent to the previous circle.Now repeat the construction: draw the circle through points Si+1, S

−i , Si, and then

draw the circle through points Si, Si−1, tangent to the previous circle. Finally,define S+

i to be the intersection point of the two “new” circles.

Proposition 5.16. The tangency of two pairs of circles meeting at point Si in

Fig. 22 is equivalent to equations (5.13)-(5.15).


Proof. A Mobius transformation sends a circle or a line to a circle or a line. Sendpoint Si to infinity; then the circles through this points become straight lines.Two circles are tangent if they make zero angle. Since Mobius transformations areconformal, the respective lines are parallel.

Thus the configuration of circles in Fig. 22 becomes a parallelogram with verticesSi−1, S

−i , Si+1 and S+

i . The quadrilateral is a parallelogram if and only if

(5.19) Si−1 + Si+1 = S−i + S+

i .

On the other hand, if Si = ∞ then equation (5.14) becomes

S+i − Si+1

S−i − Si−1

= −1,

which is equivalent to (5.19).

This circle pattern generalizes the one studied by O. Schramm in [33] in theframework of discretization of the theory of analytic functions (there the pairs ofnon-tangent neighboring circles were orthogonal). See also [4] concerning moregeneral circle patterns with constant intersection angles and their relation withdiscrete integrable systems of Toda type.

Acknowledgments. It is a pleasure to thank the Hausdorff Research Institutefor Mathematics whose hospitality the authors enjoyed in summer of 2011 where thisresearch was initiated. We are grateful to A. Bobenko, V. Fock, S. Fomin, M. Glick,R. Kedem, R. Kenyon, B. Khesin, G. Mari-Beffa, V. Ovsienko, R. Schwartz, F.Soloviev, Yu. Suris for stimulating discussions. M. G. was partially supported by theNSF grant DMS-1101462; M. S. was partially supported by the NSF grants DMS-1101369 and DMS-1362352; S. T. was partially supported by the Simons Foundationgrant No 209361 and by the NSF grant DMS-1105442; A. V. was partially supportedby the ISF grant No 162/12.

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http://arxiv.org/abs/math/0609764



Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA

E-mail address: [email protected]

Department of Mathematics, Michigan State University, East Lansing, MI 48823,

USA


Department of Mathematics, Pennsylvania State University, University Park, PA

16802, USA and ICERM, Brown University, Providence, RI 02903, USA


Department of Mathematics and Department of Computer Science, University of

Haifa, Haifa, Mount Carmel 31905, Israel


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INTEGRABLE CLUSTER DYNAMICS OF DIRECTED … · arxiv:1406.1883v1 [math.ds] 7 jun 2014 integrable cluster dynamics of directed networks and pentagram maps michael gekhtman, …